Solution
= Solution
If $X$ belongs to the <Special orthogonal Lie algebra>, which is also the Lie algebra of $O(3)$, then $e^{tX}\in O(3)$ near $t=0$. Differentiating
$$
e^{tX^{\mathsf T}}e^{tX}=I
$$
at zero gives $X^{\mathsf T}+X=0$. Conversely, the <matrix exponential> of every real <skew-symmetric matrix> is orthogonal. Hence
$$
\mathfrak o(3)=\{X\in M_3(\mathbb R):X^{\mathsf T}=-X\}.
$$