The complex spin group $Spin(n,C)$ is defined as the double cover of $SO(n,C)$.
If the the exponential map is surjective, it will give a parametrization of this Lie group. Is it true for this non-compact Lie group?
The answer is NO. See the comments for details.
I want to add one more question:
Is $SO(n,C)$ weakly exponential, i.e. is the image of the exponential map dense in $SO(n,C)$?
e.g., $SO(2,C)≅C^\times$(the punctured plane), $SO(3,C)≅PSL(2,C)$, the exponential maps of which are surjective. But for $n\geq4$ the exponential maps are not surjective.
I think the answer is useful for numerical calculations.