Let $V$ be an euclidean vector space, and $\omega \in \wedge^2 V^*$ be non degenerate, i.e. the induced homomorphism $\tilde\omega: V \to V^*$ is bijective. What is the relation between the two isomorphisms $$ L^{n-k}:\wedge^k V^* \to \wedge^{2n-k}V^*$$ (where $L$ is wedge with the Kahler form of $V$) and $$ \wedge^k V^* \cong \wedge^{2n-k} V \cong \wedge^{2n-k}V^*,$$ where the second equality is given by $\wedge^{2n-k}\tilde\omega$ and $\dim_{\mathbb R}V=2n$ ?
(This is Exercise 1.2.6 from the book Complex Geometry by Huybrechts.)