I am a bit stuck in understanding the proof of the Poincare lemma and a bit hung up on a certain part (I am following the book by Guillemin and Haine for what it is worth). The result that they are trying to prove is that on a connected open set $U \subset \mathbb{R}^n$, if $\omega \in \Omega^n_c(U)$ with $\int_U \omega = 0$, then $\omega = d \eta$ for some $\eta \in \Omega^{n-1}_c(U)$. The proof is roughly in two steps: (1) prove the result for $U$ a rectangle, and (2) use the rectangle case to prove the more general case. I am stuck on step (1).
For concreteness, I am trying to just understand the proof for $n = 2$ and to fix notation, the rectangle $R = [0,1] \times [0,1]$. Let $\omega$ be compactly supported form on $R$, and further assume that $\int_R \omega = 0$. I would like to understand how I construct a compactly-supported 1-form $\eta$ on $R$ with $d \eta = \omega$. Can someone show me how to prove this?
(For what it is worth, the relevant part of the book is Theorem 3.2.3 - I don't understand the mechanics or the motivation for the proof.)