About symplectic forms

Question

Let $(M,g)$ be a riemannian manifold and $c \in H^2(M,{\bf R})$, for which conditions does it exist $J$, a complex structure, such that $g(JX,Y)=\omega (X,Y)$ with $\omega$, a symplectic form, such that $[\omega]=c$? Or for $J$ fixed, can we find $g$?