Is the exponential map $\exp: \mathfrak{so}(3) \rightarrow \textrm{SO}(3)$ injective?

Question

Is the exponential map $\exp: \mathfrak{so}(3) \rightarrow \textrm{SO}(3)$ injective? How about the case $n>3$?

Answer 1

A compact group contains a closed subgroup isomorphic to $S^1$, and the exponential of the big group restricts to the exponential of the subgroup, which is not injective.