I am really struggling with this one: I have a two form $T=T_{i\overline{j}}dz_id\overline{z}_j$ and a Hermitian form $\omega=g_{i\overline{j}}dz_i\wedge d\overline{z_j}$. And I want to prove the formula $T\wedge \frac{\omega^{n-1}}{(n-1)!}=(g^{\overline{j}k}T_{\overline{k}j})\frac{\omega^n}{n!}$, with $g^{\overline{j}k}$ being the inverse matrix of $g_{k\overline{j}}$.
What have I tried: I tried calculating $n=1$ scenario, which reduces to $T=g^{\overline{1}1}T_{\overline{1}1}g_{\overline{1}1}dz_1d\overline{z_1}$.
However, when I tried to go to $n=2$, I have $LHS=(-T_{\overline{1}2}g_{2\overline{1}}-T_{2\overline{1}}g_{1\overline{2}}+T_{1\overline{1}}g_{2\overline{2}}+T_{2\overline{2}}g_{1\overline{1}})dz_1\wedge d\overline{z_1}\wedge dz_2\wedge d\overline{z_2}$ and $RHS=\frac{1}{2}(g^{1\overline{2}}T_{\overline{2}1}+g^{2\overline{1}}T_{\overline{1}2}+g^{1\overline{1}}T_{\overline{1}1}+g^{2\overline{2}}T_{\overline{2}2})det(g_{i\overline{j}})dz_1\wedge d\overline{z_1}\wedge dz_2\wedge d\overline{z_2}$, which I can't manage to fit together.
Now that I couldn't fit these together, I began to doubt whether the formula is true in general for Hermitian forms. Do we need the form $\omega$ to be Kahler?
Help?