Lie algebra for SO(3) as a skew symmetric matrix How can I show that the associated lie algebra for SO(3) is the set of all 3 dimensional skew-symmetric matrices?

$$
\begin{split}
\forall A\in SO(3), AA^T=I &\Rightarrow \frac{d}{dt}(AA^T)=0=\dot AA^T+A\dot A^T=\dot AA^T+(\dot AA^T)^T\\
&\Rightarrow (\dot AA^T)^T=-\dot AA^T
\end{split}
$$
and therefore $\dot AA^T$ is a $3\times 3$ skew symmetric matrix. Therefore, the Lie algebra, which is identified with the tangent space at identity:
$$
so(3):=T_eSO(3)=R_{A*}^{-1}T_ASO(3)=\{\dot A(t)A^{-1}(t)|_0|A(0)=A\}
$$
is the vector space of all $3\times 3$ skew symmetric matrices ($R_{A*}^{-1}$ is the right pull back).
