Let $L$ be a holomorphic line bundle on a complex manifold $X$, and assume it is equipped with a singular hermitian metric $h$ with local weight $\varphi$. Then, one can show that the de Rham class of $\frac{i}{\pi}\partial \overline{\partial} \varphi$ coincides with the first Chern class $c_1(L)$ of the line bundle.

Is there a generalization of this result to higher-rank vector bundles?

For example, let $E$ be a holomorphic vector bundle of rank $r$ on a complex manifold $X$ and let $h$ be a singular hermitian metric on $E$. Can we describe the Chern clases $c_1(E),\ldots,c_r(E)$ in terms of $h$ and its curvature form?