Definition. Let $\rho_V$ and $\rho_W$ be representations of a Lie group $G$ on vector spaces $V$ and $W$. Then a isomorphism of the representations if a linear bijective map $f:V\rightarrow W$ so that $$ f((\rho_V)_g v)=(\rho_W)_g f(v) $$
*Definition $2.1.3$ Hamilton's book Mathematical Gauge Theory.
Question. How can I proof (using this definition) that the following representations $$ \rho:SO(3)\rightarrow GL(\mathbb R^3) \qquad \rho_A(v)=A\cdot v$$ $$ Ad: SO(3) \rightarrow GL(\mathfrak{so}(3)) \qquad Ad_A(v)=A\cdot v \cdot A^{-1}$$ are equivalent?
First we define $f:\mathbb R^3\rightarrow \mathfrak{so}(3) $ using two basis $\{T_i\}$ and $\{e_i\}$ of $\mathfrak{so}(3)$ and $\mathbb R^3$. The obvious way to define this map is $$ f\left(\sum_i x_i e_i\right) = \sum_i x_i T_i $$
but I didn't succed to prove this directly. Is sufficient to prove $$ f(A\cdot e_i)= A \cdot f(e_i) \cdot A^{-1}$$ for each $i$ ? If so, then I got it.
