Let $\Omega(X,Y)=g(X,JY)$ is a Kaehler 2-form, and $d\Omega=\omega\wedge\Omega$ (1), where $\omega$ is closed one-form. Relation (1) is equivalent to $$(\nabla_XJ)Y=\frac{1}{2}[\Theta(Y)X-\omega(Y)JX-g(X,Y)A-\Omega(X,Y)B],$$ where $\Theta=\omega\circ J$, $g(X,B)=\omega(X)$, $A=-JB$.
I am trying to prove this equivalence, starting with $X\Omega(Z,Y)=Xg(Z,JY)$. And that is fine if I use $\nabla$ (Levi-Civita connection), for example $\nabla_X\Omega(Z,Y)+\Omega(\nabla_XZ,Y)+...$. But, how will I use $d$ in this case?