Are there holomorphic vector bundles that do not holomorphically embed in any free one?

Question

Are there any examples of holomorphic vector bundles $E \to M$ of rank $k$ that do not admit a holomorphic classifying map $M \to \mathrm{Gr}_k(\mathbb C^n)$ for any $n \in \mathbb N$? Or, equivalently, holomorphic vector bundles $E \to M$ that do not holomorphically embed in any free one.

Answer 1

The line bundle $\mathcal{O}(1)$ does not embed into a trivial vector bundle, and in fact $\mathrm{Hom}(\mathcal{O}(1),\mathcal{O}) = 0$.