Suppose we have a general Lie group $G$ and it's corresponding Lie Algebra $\mathfrak{g}$. I think that it is true that for all $X \in \mathfrak{g}$ we have $\mathrm{exp}(X) \in G$.
However, the converse is not true because I don't think $\mathrm{exp}$ will cover all of $G$. Meaning, if I hand you $g \in G$, you cannot tell me with certainty that there exists some $X \in \mathfrak{g}$ such that $\exp(X) = g$ (in general).
I think though, that $SO(3)$ $does$ have this converse property. That is to say I think it is true that $\forall A \in SO(3)$, $\exists X \in \mathfrak{so}(3)$ such that $A = \mathrm{exp}(X)$.
First of all, am I right in thinking this? Physical intuition say that it is true.
Secondly, if it is true is there some way to prove this? Or is there some kind of theorem about this result?