Recall that any $f \in C^\infty(\mathbb{T}^n)$ can be uniquely written as a convergent Fourier series
$$
f = \sum_{\mathbb{k} \in \mathbb{Z}^n} f_{\mathbb{k}} U_{\mathbb{k}}, \quad f_{\mathbb{k}} := \int_{\mathbb{T}^n} e^{-2\pi i \langle \mathbb{k},t \rangle}f(t)\,dt,
$$
where for each $\mathbb{k} \in \mathbb{Z}^n$,
$$
\forall t \in \mathbb{T}^n, \quad U_{\mathbb{k}}(t) := e^{2\pi i \langle \mathbb{k},t\rangle}.
$$
Recall, moreover, that the usual pointwise multiplication of functions corresponds to the convolution product of Fourier series:
$$
\forall f, \; g \in C^\infty(\mathbb{T}^n), \quad fg = \sum_{\mathbb{k} \in \mathbb{Z}} \left(\sum_{\mathbb{k}^\prime \in \mathbb{Z}^n} f_{\mathbb{k}-\mathbb{k}^\prime}g_{\mathbb{k}^\prime} \right)U_{\mathbb{k}}.
$$
In other words, $C^\infty(\mathbb{T}^n)$ is generated by unitaries $\{U_{\mathbb{k}}\}_{\mathbb{k} \in \mathbb{Z}^n}$ satisfying the relations
$$
\forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad U_{\mathbb{k}}U_{\mathbb{k}^\prime} = U_{\mathbb{k}+\mathbb{k}^\prime},
$$
which implies, in turn, that
$$
\forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad U_{\mathbb{k}^\prime}U_{\mathbb{k}} = U_{\mathbb{k}}U_{\mathbb{k}^\prime}.
$$
Now, suppose that $\Theta : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{T} := \mathbb{R}/\mathbb{Z}$ is a (normalised) $2$-cocycle, i.e., that it satisfies
- For all $\mathbb{k}\in\mathbb{Z}^n$, $\Theta(\mathbb{0},\mathbb{k}) = \Theta(\mathbb{k},\mathbb{0}) = 0$ (normalisation).
- For all $\mathbb{p}$, $\mathbb{q}$, $\mathbb{r} \in \mathbb{Z}^n$, $\Theta(\mathbb{p},\mathbb{q}+\mathbb{r}) + \Theta(\mathbb{q},\mathbb{r}) = \Theta(\mathbb{p},\mathbb{q}) + \Theta(\mathbb{p}+\mathbb{q},\mathbb{r})$ ($2$-cocycle condition).
Then you can replace the usual convolution of Fourier series by a deformed convolution of Fourier series to obtain the noncommutative $n$-torus $C^\infty(\mathbb{T}^n_\Theta) := (C^\infty(\mathbb{T}^n),\star_\Theta)$:
$$
\forall f,g \in C^\infty(\mathbb{T}^n), \quad f \star_\Theta g := \sum_{\mathbb{k} \in \mathbb{Z}} \left(\sum_{\mathbb{k}^\prime \in \mathbb{Z}^n} e^{-2\pi i \Theta(\mathbb{k}-\mathbb{k}^\prime,\mathbb{k}^\prime)}f_{\mathbb{k}-\mathbb{k}^\prime}g_{\mathbb{k}^\prime} \right)U_{\mathbb{k}}.
$$
In terms of our unitary generators $\{U_{\mathbb{k}}\}_{\mathbb{k} \in \mathbb{Z}^n}$, this reduces to defining
$$
\forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad U_{\mathbb{k}} \star_\Theta U_{\mathbb{k}^\prime} := e^{-2\pi i \Theta(\mathbb{k},\mathbb{k}^\prime)}U_{\mathbb{k}+\mathbb{k}^\prime},
$$
which, in turn, implies the commutation relations
$$
\forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad U_{\mathbb{k}^\prime} \star_\Theta U_{\mathbb{k}} = e^{2\pi i \theta(\mathbb{k},\mathbb{k}^\prime)}U_{\mathbb{k}} \star_\Theta U_{\mathbb{k}^\prime},
$$
where $\theta : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{T}$ is defined by
$$
\forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad \theta(\mathbb{k},\mathbb{k}^\prime) := \Theta(\mathbb{k},\mathbb{k}^\prime) - \Theta(\mathbb{k}^\prime,\mathbb{k}).
$$
As it turns out, $\theta$ is an alternating bicharacter, i.e.,
- For all $\mathbb{k} \in \mathbb{Z}^n$, $\theta(\mathbb{k},\mathbb{k}) = 0$ (alternating).
- For all $\mathbb{k} \in \mathbb{Z}^n$, $\mathbb{k}^\prime \mapsto \theta(\mathbb{k},\mathbb{k}^\prime)$ and $\mathbb{k}^\prime \mapsto \theta(\mathbb{k}^\prime,\mathbb{k})$ both define homomorphisms $\mathbb{Z}^n \to \mathbb{T}$ (bicharacter).
Conversely, every alternating bicharacter $\theta : \mathbb{Z}^n \to \mathbb{Z}^n$ can be induced in this way from a (normalised) $2$-cocycle $\Theta : \mathbb{Z}^n \to \mathbb{Z}^n$.
Now, what happens when $\Theta$ and $\Theta^\prime$ both induce the same alternating bicharacter $\theta$? It's an old theorem of Kleppner's that this happens if and only if $\Theta$ and $\Theta^\prime$ are cohomologous, i.e., there exists some $T : \mathbb{Z}^n \to \mathbb{T}$ such that
$$
\forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad \Theta^\prime(\mathbb{k},\mathbb{k}^\prime) = \Theta(\mathbb{k},\mathbb{k}^\prime) + T(\mathbb{k}) + T(\mathbb{k}^\prime) - T(\mathbb{k}+\mathbb{k}^\prime) ;
$$
in other words, $\Theta^\prime - \Theta = dT$. In that case, we can define an explicit $\mathbb{T}^n$-equivariant isomorphism $\Psi_T : C^\infty(\mathbb{T}^n_\Theta) \to C^\infty(\mathbb{T}^n_{\Theta^\prime})$ by
$$
\forall \mathbb{k} \in \mathbb{Z}^n, \quad \Psi_T(U_{\mathbb{k}}) = e^{2\pi i T(\mathbb{k})} U_{\mathbb{k}}.
$$
Hence, up to equivariant isomorphism, for any alternating bicharacter $\theta$ (or equivalently, by Kleppner, for any class $\theta \in H^2(\mathbb{Z}^n,\mathbb{T})$, where $H^2(\mathbb{Z}^n,\mathbb{T})$ is the second group cohomology of $\mathbb{Z}^n$ with coefficients in $\mathbb{T}$), we can define the noncommutative $n$-torus $C^\infty(\mathbb{T}^n_\theta)$ as the algebra generated by unitaries $\{U_{\mathbb{k}}\}_{\mathbb{k} \in \mathbb{Z}^n}$ satisfying the commutation relations
$$
\forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad U_{\mathbb{k}^\prime} U_{\mathbb{k}} = e^{2\pi i \theta(\mathbb{k},\mathbb{k}^\prime)}U_{\mathbb{k}} U_{\mathbb{k}^\prime};
$$
to construct it, we just take $C^\infty(\mathbb{T}^n_\theta) := C^\infty(\mathbb{T}^n_\Theta)$ for any $2$-cocycle $\Theta$ inducing $\theta$.
Let me be more explicit about what appears in the literature. We have an isomorphism
$$
\{\text{alternating bicharacters $\theta : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{T}$}\} \cong \mathbb{T}^{n(n-1)/2}
$$
given by
$$
\theta \mapsto (\theta(e_i,e_j))_{1 \leq i < j \leq n},
$$
where $\{e_i\}_{i=1}^n$ is the standard ordered basis for $\mathbb{R}^n$; conversely, the alteranting bicharacter $\theta$ corresponding to $(\theta_{ij})_{1\leq i < j \leq n} \in \mathbb{T}^{n(n-1)/2}$ is given by
$$
\forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad \theta(\mathbb{k},\mathbb{k}^\prime) = \sum_{1 \leq i < j \leq n} \theta_{ij}(k_i k^\prime_j - k_j k^\prime_i).
$$
Then there are three conventions that tend to appear for constructing $2$-cocycles $\Theta : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{T}$ inducing $\theta$:
- For all $\mathbb{k}$, $\mathbb{k}^\prime \in \mathbb{Z}^n$, set $\Theta(\mathbb{k},\mathbb{k}^\prime) := \sum_{1 \leq i < j \leq n} \theta_{ij} k_i k^\prime_j$.
- For all $\mathbb{k}$, $\mathbb{k}^\prime \in \mathbb{Z}^n$, set $\Theta(\mathbb{k},\mathbb{k}^\prime) := -\sum_{1 \leq i < j \leq n} \theta_{ij} k_j k^\prime_i$.
- For each $1 \leq i < j \leq n$, choose $\tfrac{\theta_{ij}}{2}$ such that $2 \tfrac{\theta_{ij}}{2} = \theta_{ij}$. Hence, for all $\mathbb{k}$, $\mathbb{k}^\prime \in \mathbb{Z}^n$, set $\Theta(\mathbb{k},\mathbb{k}^\prime) := \sum_{1\leq i < j \leq n}\tfrac{\theta_{ij}}{2}(k_i k^\prime_j - k_j k^\prime_i)$.
The first two conventions give you non-alternating bicharacters, which complicates a fair bit of the algebra, but turn out to be far more robust from a group-cohomological perspective; in particular, they give honest-to-goodness splittings of the short exact sequence
$$
0 \to \ker(\Theta \mapsto \theta) \to \{\text{$2$-cocycles}\} \xrightarrow{\Theta \mapsto \theta} \{\text{alternating bicharacters}\} \to 0.
$$
The third convention gives you alternating bicharacters, which simplifies a fair bit of the algebra, but all the non-uniqueness can give you headaches when dealing with certain technicalities.
