I want to solve the following exercise:
Let $\omega = \frac{i}{2\pi} \sum dz_i \wedge d\overline{z_i}$ be the standard fundamental form on $\mathbb{C}^n$. Show that one can write $\omega = \frac{i}{2\pi}\partial\overline{\partial}\varphi$ for some positive function $\varphi$ and determine $\varphi$.
Well, now that's my idea: $\frac{i}{2\pi} \partial \overline{\partial} \varphi = \frac{i}{2\pi} \partial (\sum_{k=1}^n \frac{\partial \varphi}{\partial \overline{z_k}} d\overline{z_k}) = \frac{i}{2\pi} \sum_{l=1}^n \frac{\partial}{\partial z_l} (\sum_{k=1}^n \frac{\partial \varphi}{\partial \overline{z_k}} d\overline{z_k}) dz_l$. That's just the definition, isn't it? Or do I somehow need to use a chain rule or something like that to calculate $\partial \overline{\partial}$?
If that's right what I did until now, how do I get the wedge product into my formula? What is the connection between wedge product and $\partial \overline{\partial}$-operator?
Thanks in advance!