For the unitary group $U(n)$, its group elements are of the form $e^{iH}$, with $H$ being a $n\times n $ hermitian matrix.
Do we have a similar expression for the group element of the symplectic group $Sp(2n,\mathbb{C})$? The group is defined as
$$ Sp(2n,\mathbb{C}) =\{ A \in M_{2n\times 2n } (\mathbb{C}), A \Omega A^T = \Omega \}, $$
where $\Omega$ is defined as
$$ \Omega = \left( \begin{array}{cc} 0 & I_n \\ -I_n & 0 \end{array} \right) . $$
Yes, we have $A = e^B \in \mathrm{Sp}(2n,\mathbb{C})$, with $B \in \mathfrak{sp}(2n,\mathbb{C}) = \{M \in \mathrm{Mat}(2n,\mathbb{C}) \;|\; \Omega M + M^T \Omega = 0\}$, the latter being the Lie algebra associated to the Lie group of complex symplectic matrices.