2-forms represented by a first Chern class?

Question

Let $M$ be a complex manifold and $\omega$ be a 2-form on $M$. Is there a good way to see whether $\omega$ is represented by the first Chern class of a line bundle on $M$? In other words, when is it of the form $\omega=2\pi i c_1(L)$ for a line bundle $L$ on $M$?