Here is an explicit procedure based on the isomorphism between the de-Rham and Cech cohomologies
for smooth manifolds based on R. Bott and L.W. Tu's book: Differential forms in algebraic topology.

The description will be given for a three form but it can be generalized
along the same lines to forms of any degree. We suppose that the manifold has a finite good cover.

The data needed is the transition functions between the coordinate charts (which will be denoted by: $U_\alpha$, $U_\beta$, etc.)
and of course, the coordinate expression of the given form on each chart.

Given a three form F, then by the Poincare lemma, on $U_\alpha, F = dB_\alpha$, where $B_\alpha$ are two forms on $U_\alpha$.

Thus by the Poincare lemma on $U_\alpha \cap U_\beta$:

$B_\alpha-B_\beta = dA_{\alpha\beta}$ , where $A_{\alpha\beta}$ are one forms on $U_\alpha \cap U_\beta$.

Since on $U_\alpha \cap U_\beta \cap U_\gamma$:

$d(A_{\alpha\beta}+A_{\beta\gamma}+A_{\gamma\alpha})=0$

Then by the Poincare lemma

$A_{\alpha\beta}+A_{\beta\gamma}+A_{\gamma\alpha} = d\phi_{\alpha\beta\gamma}$

where: $\phi_{\alpha\beta\gamma}$ are zero forms on $U_\alpha \cap U_\beta \cap U_\gamma$.

Again, Since on: $U_\alpha \cap U_\beta \cap U_\gamma \cap U_\delta$:

$d(\phi_{\alpha\beta\gamma}-\phi_{\beta\gamma\delta}+\phi_{\gamma\delta\alpha}-\phi_{\delta\alpha\beta})=0$

Then:

$\phi_{\alpha\beta\gamma}-\phi_{\beta\gamma\delta}+\phi_{\gamma\delta\alpha}-\phi_{\delta\alpha\beta} = C_{\alpha\beta\gamma\delta}$

where: $C_{\alpha\beta\gamma\delta}$ are constants.

The differential form F is integral iff:

$C_{\alpha\beta\gamma\delta}= 2 \pi n_{\alpha\beta\gamma\delta}$

where $n_{\alpha\beta\gamma\delta}$ are integers on all quadruple intersections.