I am having a hard time wrapping the exponential map around my head. So if $G$ is a Lie group we have this map :
$exp : T_eG \rightarrow G $
My understanding of lie groups is that they represent stuff like rotations. So if I take $G=O(n)$ the rotations, then I know that $T_{I_n}O(n) = \text{skew symmetric matrices}$.
So I take one skew symmetric matrix $S$ which as I understand it somehow represents an infinitesimal rotation. Then $exp(tS)$ is a curve for which the elements are rotations of $O(n)$. But what do these element mean? How to think intuitively about what this exponential map actually is?
Edit :
Sorry, but I don't see how the linked answers tell me how to think about the curve $\gamma(t)=exp(tX)$, and neither did OP since apparently they were not accepted...