Let $S$ be a connected orientable topological compact surface of genus $g$. There are various ways to prove that the second singular homology group with integer coefficients $H_2(S, \mathbb{Z})$ is a free abelian group of rank $1$.
This question concerns how to concretely write down a generator using the definition of singular homology. That is, we need to find a singular $2$-chain with zero boundary.
The idea would be of course to use a triangulation of $S$ and choose suitable signs on each singular $2$-simplex so that the corresponding chain has no boundary. But when I try to make this precise I run into the problem that one oriented face of a triangle minus the same face with the reversed orientation is not the zero $1$-chain.
Is there some tricky way to choose the signs that makes things work (for example, representing $S$ as the quotient of a $4g$-gon and considering a triangulation out of its center) or one is forced to use that singular homology is isomorphic to simplicial or cellular homology?
I would be very grateful if someone knew how to do that, or a reference where it is properly explained.