There are lots of references. Mainly every textbook which treats Hodge theory. Try to look at:
- Voisin: Hodge theory and complex algebraic geometry. I
- Huybrechts: Complex geometry
- Wells: Differential analysis on complex manifolds
- Griffiths, Harris: Principles of algebraic geometry
There, you will find mainly the proof in the case $n=2$, which is used to prove the Lefschez theorem on $(1,1)$-classes. The general case is a straightforward adaptation of that argument.
