On the way to homology cross product

Question

Let X and Y be topological spaces and let $\sigma_m : \Delta^m \rightarrow X$ and $\sigma_n: \Delta^n \rightarrow Y$ be continuous maps from the standard simplices of dimensions $m$ and $n$ respectively into X and Y (i.e., two singular simplices of X and Y). We have the product map $$\sigma_m \times \sigma_n: \Delta^m \times \Delta^n \rightarrow X \times Y;$$ I want to write (?) this product map as a sum of $\binom{m+n}{m}$ singular $(m+n)$-simplices. I want to use the fact that $\Delta^m \times \Delta^n$ can be triangulated into $\binom{m+n}{m}$ simplices of dimension $m+n$. I don't know how to write the product map on each simplex. Thanks for the help.