Motivated by this question, I began to wonder if there is a global definition of the almost complex structure of a complex manifold. It is (almost) always presented as multiplication by complex $i$ on the tangent space, and then globalized. Using the formulae given earlier $$ \overline{\partial}\omega = \frac{1}{2}(\text{d}\omega + i \text{d}(J\omega)), $$ and $$ \partial \omega = \frac{1}{2}(\text{d}\omega - i \text{d}(J\omega)), $$ it is easy to see that $$ -\frac{i}{2}d\omega = d(J\omega). $$ Thus, $J\omega = -\frac{i}{2}\omega + \omega'$, where $\omega'$ is some closed form. What this $\omega'$ is, however, I cannot see.