Question

Let $M$ be a complex manifold, with a Hermitian metric $g$ which we will consider as a $ C^\infty(M)$-bi-module map $$ g:\Omega^1(M) \otimes_{C^{\infty}(M)} \Omega^1(M) \to C^\infty(M), $$ where $\Omega^1(M)$ is the module of one forms of $M$. Moreover, let $\omega$ be the fundamental form of $g$. Now contraction by $\omega$ is a map $$ \omega \llcorner :\Omega^1(M) \wedge \Omega^1(M) \to C^\infty(M). $$ Am I naive in thinking that there might be some simple relationship between these two maps? For example, $$ (\omega \llcorner) \circ \wedge = g, $$ where $$ \wedge:\Omega^1(M) \otimes_{C^{\infty}(M)} \Omega^1(M) \to \Omega^1(M) \wedge \Omega^1(M) $$ is the obvious map. Does it help if I assume that $g$ is Kaehler, ie d$\omega = 0$?