Consider a Hermitian metric $h$ on a complex vector bundle. We know that we can form a Riemannian metric $g$ and closed $(1,1)$ curvature form $\omega$ such that
$$ h = g -i\omega $$ where $$ g = \frac{1}{2}(h+\bar{h}) $$
$$ \omega = \frac{i}{2}(h-\bar{h}) $$
My question is, assuming the bundle is topologically trivial so as to allow $\omega = \text{d}A$ for some connection $A$, is $A$ metric compatible with the Riemannian metric above?