Of course, one can define real vector spaces $\Bbb R^n$ for any positive integer $n$. What distinguishes the real numbers ($\Bbb R$), the complex numbers ($\Bbb C$), quaternions ($\Bbb H$), and octonions $(\Bbb O)$ is additional structure, including a multiplication operation $\Bbb A \times \Bbb A \to \Bbb A$.
Hurwitz' Theorem states that (up to isomorphism) these are precisely all of the normed division algebras over $\Bbb R$. These are algebras over $\Bbb R$ with (1) the property that $x y = 0$ implies $x = 0$ or $y = 0$ and (2) a multiplicative norm $||\,\cdot\,||$ (so, satisfying $||x y|| \leq ||x||\,||y||$). (For a truly enjoyable introduction to this rich circle of ideas, I highly recommend John Baez' classic article The Octonions.)
One can still define, however, other canonical algebraic operations $\Bbb A \times \Bbb A \to \Bbb A$ on real vector spaces $\Bbb A = \Bbb R^n$ for various small $n$ that still have other interesting properties, and the normed division algebras and the Cayley-Dickson construction lead naturally to many of these.
For example, each of the normed division algebras is equipped with an (involutive) linear conjugation operation $\bar\cdot : \Bbb A \to \Bbb A$ compatible with the addition, multiplication, and norm operations (for $\Bbb R$ this is just the trivial map, for $\Bbb C, \Bbb H, \Bbb O$ these are just the usual conjugation maps). In each case, the $+1$-eigenspace of the conjugation map is just the copy of $\Bbb R$, and so the $-1$-eigenspace has dimension $\dim \Bbb A - 1$. We denote the latter space $\operatorname{Im} \Bbb A$ (so, as a vector space, it is isomorphic to $\Bbb R^{\dim \Bbb A - 1}$), denote the projection $\Bbb A \to \operatorname{Im} \Bbb A$ onto it by $\operatorname{Im}$, and call its elements imaginary. Now, $\Bbb A \times \Bbb A \to \operatorname{Im} \Bbb A$, $(x, y) \mapsto \operatorname{Im} (x \bar y)$, restricts to an new, skew-symmetric binary operation, $$\times:\operatorname{Im} \Bbb A \times \operatorname{Im} \Bbb A \to \operatorname{Im} \Bbb A .$$ In particular, these operations do not have identities (after all, the identity $1 \in \Bbb A$ is not imaginary).
For $\Bbb A = \Bbb R$ and $\Bbb A = \Bbb C$, this construction just leads to the zero map on $\Bbb R^0$ and $\Bbb R^1$---so, not very interesting.
For $\Bbb A = \Bbb H$, $\times : \Bbb R^3 \times \Bbb R^3 \to \Bbb R^3$ is nothing other than the familiar cross product, recovering Henning Makholm's suggestion. Tracing through definitions (and doing some easy calculations) shows that associativity of $\Bbb H$ implies the triple cross product identity $({\bf x} \times {\bf y}) \times {\bf z} = \langle {\bf y}, {\bf z} \rangle {\bf x} - \langle {\bf x}, {\bf z} \rangle {\bf y}$ on $\Bbb R^3$, and forming the cyclic sum of this identity in ${\bf x}, {\bf y}, {\bf z}$ gives the Jacobi identity, $$({\bf x} \times {\bf y}) \times {\bf z} + ({\bf y} \times {\bf z}) \times {\bf x} + ({\bf z} \times {\bf x}) \times {\bf y} = {\bf 0} ,$$
so $\times$ is actually the bracket operation of a real Lie algebra, namely $\mathfrak{so}(3, \Bbb R)$. (NB that $\times$ itself is not associative.) More generally, any property of the multiplication on $\Bbb A$ induces a property of $\times$ on $\operatorname{Im} \Bbb A$.
For $\Bbb A = \Bbb O$, $\times : \Bbb R^7 \times \Bbb R^7 \to \Bbb R^7$ is the somewhat exotic $7$-dimensional cross product. It does not satisfy the Jacobi identity, so it is not the bracket of some Lie algebra, but it is intimately related to (the compact real form) of the exceptional Lie algebra $\mathfrak{g}_2$.
We can generate more examples of canonical algebraic structures on $\Bbb R^n$ if we tweak the Cayley-Dickson construction by defining for a suitable algebra $\Bbb A$ the multiplication rule $$(a, b)(c, d) := (ac \color{red}{+} \bar d, da + b \bar c)$$ on $\Bbb A \times \Bbb A$ (replacing the red $\color{red}{+}$ with $-$ gives the usual Cayley-Dickson construction). If we successively apply this again to $\Bbb R$, we produce
the split-complex numbers ($\widetilde {\Bbb C}$), the split-quaternions ($\widetilde {\Bbb H}$), and the split-octonions ($\widetilde {\Bbb O}$), again respectively of dimension $2, 4, 8$. These are not normed division algebras---instead of a norm, these are equipped with an indefinite quadratic form, making them composition algebras (and together with the four normed division algebras, this accounts for all real composition algebras up to isomorphism). These algebras, as the name and notation suggest, share many algebraic features with their nonsplit counterparts. Two are actually familiar: As $\Bbb R$-algebras, $\widetilde{\Bbb C} \cong \Bbb R \oplus \Bbb R$ and $\widetilde{\Bbb H} \cong M(2, \Bbb R)$ (the space of $2 \times 2$ real matrices). In particular, unlike the normed division algebras, these algebras have zero divisors.
The same construction as before now leads to a "split" cross product on $\Bbb R^3$ that can be realized as the Lie bracket on $\mathfrak{sl}(2, \Bbb R) \cong \mathfrak{so}(1, 2)$, as well as a "split" cross product on $\Bbb R^7$. In both cases these are inequivalent to the cross products described earlier.
Finally, something special happens in the split setting that has no analogue in the normed division algebra setting: Up to isomorphism, there is a $6$-dimensional algebra $\Bbb S \cong \Bbb R^6$, $\widetilde{\Bbb H} \subset \Bbb S \subset \widetilde{\Bbb O}$, unique up to isomorphism called, of course, the sextonions. It turns out (but is not immediately obvious) that the split $7$-dimensional cross product restricts to a binary operation on $\operatorname{Im} \Bbb S \cong \Bbb R^5$.
