Setup: A smooth manifold $M$, and a locally finite contractible open cover $\mathscr{U}$ with a subordinate partition of unity $\{h_i\}$.
Define a Čech cohomology $H^n(M, \mathscr{U})$ in the usual manner, where the $p$-cochains $\mathscr{F}_p=\{f_{ij...k}\}$ are collections of locally constant functions.
The de Rham cohomology is isomorphic to the Čech cohomology, and it is known that we can write $\omega=f_{ijk}h_idh_j\wedge dh_k$ (there is a sum over the repeated indices) for, say, $[\omega]\in H^2_{dR}(M)$.
Claim: Let $[\omega]$ be an integral class in the de Rham cohomology. Then, $\mathrm{Im}(f_{ijk})\subseteq \mathbb{Z}$.
Proof: The obvious thing to do is try integrating $f_{ijk}h_idh_j\wedge dh_k$, I suppose? We know that that will be an integer; and also that it is only non-zero inside $U_i \cap U_j \cap U_k$. But I'm not sure how to conclude the result from here. Maybe if we could arrange our $2$-chains such that the integral ended up being $f_{ijk}$ evaluated at a point? Help appreciated...