(1) As I read some article in here ( I cannot found ), so we know that $$ {\rm exp} \ (T_eSO(3)) \neq SO(3) $$
( ${\rm diag}(-1,-1,1)$ cannot be covered by ${\rm exp}$ )
But there exists some open ball $B$ in $T_eSO(3)$ so that ${\rm exp}\ B$ is open.
Then can we choose $\{ g_1, ... , g_n \}\subset SO(3)$ such that $\{ g_i B\}$ is open cover ? If so, can we find smallest $n$ ?
(2) My second question is : What is ${\rm exp}\ (T_eSO(3))$ ?
$$\{ {\rm exp}\ tE_{12} | t\in {\bf R} \}=SO(2)$$ where $E_{12}$ is a matrix having only nonzero $(1,2),\ (2,1)$-entries which are $1$ and $-1$. That is $\{ {\rm exp}\ tE_{12} | t\in {\bf R} \}$ is a closed geosic in $SO(3)$. But what is ${\rm exp}\ (T_eSO(3))$ ?
Rethink your assumptions - $\rm diag(-1,-1,1)$ is the exponential of $$\left(\begin{matrix} 0& \pi& 0\\ -\pi& 0& 0\\ 0& 0& 0\end{matrix}\right).$$ In fact the exponential map is surjective - this is true for any compact connected Lie group.