Assume that $U$ is an open subset of $\mathbb{C}^n$. We fix the standard almost complex structure $J$ on $U$.
Assume that $\omega$ is an arbitrary symplectic structure on $U$.

Is there a Riemannian metric on $U$ with the following property?

>For every smooth function $f: U \to \mathbb{R}$,   we  have  $J \nabla f = X_f$  where  $X_f$  is  the  Hamiltonian vector  field  corresponding to $f$