Is this differential form $\overline{\partial}$ exact?

Question

1. Let $(z,w)$ be ccordinates on $\mathbb C^2$. It is not difficult to show that $1$ - form $$\omega:=\dfrac{\overline{zw} \cdot\overline{dw}-\overline{w}^2\cdot\overline{dz}}{(|z|^2+|w|^2)^3}$$

is $\overline{\partial}$ closed on $\mathbb C^2\backslash0$.

2. Is it true that it is exact?
3. Can I find explicitly smooth function $f$, such that $\omega = \overline{\partial}f$?