In my homework problem I have to prove that
$F: (\Bbb{R}^3,\times) \to (so(3),[,]),\ F(v)=\begin{pmatrix}0&-v_3&v_2 \\ v_3& 0&-v_1\\ -v_2&v_1&0 \end{pmatrix}=\hat{v}$ is a homeomorphism of Lie Algebras. Furthermore, for $w \in \Bbb{R}^3$ we have $\hat{v}w=v \times w$.
I have proved the above facts. The last part of the problem sounds like this:
Prove that for $A \in O(3)$ we have $Ad_A \hat{v}=\widehat{Av}$, i.e. $A\hat v A^{-1}=\widehat{Av}$. (where taking "$\text{^}$" means applying F to the corresponding vector)
Is there an easy way to prove this last part, without brute force computations?