I want to find the Lie algebra of $O(n)$. I know $Lie(O(n))=\{X \in \mathfrak{gl}_n(\mathbb{R}) : \exp(tX)\in O(n) \forall t\in \mathbb{R}\}$
So $\exp(tX)^T \exp(tX)=I_n$ i.e. $\exp(tX^T)\exp(tX)=I_n$ then it seems that $tX^T$ and $tX$ don't commute in general so I don't know (this is the point where I need some help) if I can say $\exp(tX^T+tX)=\exp(0)$ then take $t$ small enough such that $tX\in U$ such that $\exp|_U$ is a diffeomorphism so in particular injective. Then $tX^T+tX=0$ so $X^T+X=0$.