Let's take the tangent bundle $TM\rightarrow M$(not to be Kahler), and the first Chern class $$c_1(M)=[tr(\frac{\sqrt -1}{2\pi}\Omega)]$$ We know that the inside trace of the curvature form is a closed form.
But I try to write the form in the bracket more explicitly $$tr(\Omega_i^{j})=\Omega_i^{i}=R^{i}_{ik\bar l}dz^k\wedge dz^{\bar l}=Ric_{k\bar l}dz^k\wedge dz^{\bar l}$$
It seems that the first Chern class $c_1(M)$ can be represented by $\frac{\sqrt -1}{2\pi}[Ric]$（I know this is true for Kahler manifold, since the Ricci form $\partial_i \bar{\partial_j}log (detg) dz^i\wedge dz^{\bar j}$ is closed by using Kahler condition, but I cannot guarantee the Ricci form for any complex manifold is still a closed one.
What am I missing?