By Daniel Huybrechts in Complex Geometry, we have:
For any holomorphic vector bundle $E$ of rank $r$ there exists a non-degenerate pairing $$\bigwedge^{k}E \times \bigwedge^{r-k}E \longrightarrow \text{det}(E)$$ that induces a isomorphism of holomorphic vector bundles $$\bigwedge^{k}E \simeq \bigwedge^{r-k}E^{*} \otimes \text{det}(E)$$
Can someone give me a reference where I find the demonstration of this result?
Any help will be welcome.
Thanks a lot.