 Fundamental form of a Hermitian manifold

For the underlying Riemannian metric $g$ and complex structure $J$, the fundamental form is $\omega (u,v)=g(Ju,v)$. It is a real (1, 1)-form. In complex dimension $n$, the compatible orientation has Riemannian volume form ${\omega }^n/n!$, and its Hodge star operator satisfies