I'm not extremely familiar with algebraic geometry, but I don't see what the problem is ? And yes that's how it's proven in the differential case (cf. my original post) but perhaps there is a holomorphic equivalent.

After a bit of thinking, i'm pretty sure hat I'm asking is the following : Let $E \rightarrow X$ be a holomorphic bundle over a Kähler manifold $X$, of rank $r$. Can we find an integer $N$ and a holomorphic map $f : X \rightarrow G_r(\mathbb{C}^N)$ Such that $E = f^*\mathcal{O}$, where $\mathcal{O}$ would be the tautological bundle.