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Let $X$ be a compact connected Riemann surface and let $omega$ be a two-form on $X$. We can view the cohomology class $[omega]$ as an element of the Čech cohomology group $check{H}^2(X,mathbb{R})$, represented by some real numbers $omega_{alphabetagamma}inmathbb{R}$ on triple intersections of a cover ${U_alpha}$.

Is there an explicit formula for the integral $int_Xomega$ in terms of the real numbers $omega_{alphabetagamma}$?

For any covering $mathcal U = {U_alpha}_{alphain A}$ of $X$, we can form its Cech nerve $C(mathcal U)$, an abstract simplicial complex with vertex set $A$ whose simplices are exactly the finite subsets $Isubset A$ such that $bigcap_{alphain I} U_alphaneqemptyset$. We can then glue standard simplices together as specified by this abstract simplicial complex to get a CW complex $|C(mathcal U)|$. If $mathcal U$ is a good cover, i.e. all finite intersections are empty or contractible, and $X$ is sufficiently nice, e.g. a manifold, there is a homotopy equivalence $|C(mathcal U)|to M$ constructed by mapping the $0$-simplex corresponding to $alpha$ to some point of $U_alpha$ and then iteratively extending it over simplices (which is always possible and unique up to homotopy by contractibility). The cellular complex of $|C(mathcal U)|$ is precisely the Cech complex of $U$. Thus the Cech cocycle $omega_{alphabetagamma}$ is just a representative of the pullback of $omega$ in the cellular complex. Integration is pairing against the fundamental class, which is an element of $H_2(X)cong H_2(|C(mathcal U)|)$, so it can also be expressed as a sum of formal elements corresponding to nonempty intersections of $U_alpha$'s.

In fact, for special coverings one can identify this class explicitly: Choose a triangulation of $X$, i.e. an abstract simplicial complex $K$ and a homeomorphism $|K|cong X$. To every vertex $x$ of $K$ corresponds a contractible open subset of $|K|$, the star of $x$, which is the union of the interiors of all simplices which have $x$ as a vertex (together with $x$ itself). Together, these form a good cover of $|K|$, and the corresponding Cech nerve is just $K$ itself. The fundamental class is given as the sum of all top-dimensional (in this case $2$-dimensional) simplices, with signs corresponding to the orientations. Thus pairing with the fundamental class sends a Cech cocycle ${omega_{xyz}}$ to the sum over all triangles $t$ of $K$, with vertices $x_t,y_t,z_t$ in this order, of $omega_{x_t,y_t,z_t}$.