Expression for the form $F^k\wedge \omega^{n-k}$

Question

Let $E\to X$ be a holomorphic Hermitian vector bundle over a compact Kahler manifold. Let us write the curvature as follows $$F=\sum_{i,j}dz_i\wedge d\bar{z}_j\otimes f_{ij}$$ where $f_{ij}\in End(E)$ such that $f_{ij}^*=f_{ji}$. The Kahler form writes $$\omega=i\sum_jdz_j\wedge d\bar{z}_j.$$ I am trying to find a formula for the form $$F^k\wedge \omega^{n-k}\in\Omega^{n,n}(X,End(E)),$$ in other word an endomorphism $g\in End(E)$ such that $$F^k\wedge \omega^{n-k}=\omega^n\otimes g.$$ In particular I want to know whether $g^*=g$.