A complex symplectic space is a pair $(V, \omega)$, where $V$ is a complex vector space of complex dimension $2n$ and $\omega$ is a non-degenerate skew-symmetric $\mathbb{C}$-bilinear form. My question is:
Can we define a Hermitian inner product $H(-,-)$ on $V$ out of the complex symplectic form $\omega(-,-)$ in a similar fashion to case of a real symplectic space $W$ together with a compatible complex structure $J: W\rightarrow W$?