I am looking to do some concrete computations of (co)homology cross products with some explicit simplicial complexes. Let $K$ and $L$ be to finite abstract simplicial complexes. I have seen the definition of the cross product for singular (co)homology and I have seen how if one chooses explicit Eilenberg-Zilber chain maps then one can get an explicit formula for the singular cross product. I would like to know how to do this simplicially.
I know that there is a nice way to form a abstract simplicial complex $K \otimes L$ such that $| K \otimes L | \cong |K| \times |L|$. I imagine that there are some nice simplicial Eilenberg-Zilber maps $$ S(K) \otimes S(L) \to S(K \otimes L) \\ S(K \otimes L) \to S(K) \otimes S(L) $$ and if I new these this would give me the definition of the simplicial cross product in (co)hmology.
I am fairly new to things so I could be thinking of this all wrong. Is this the correct idea to define a simplicial cross product and if so how can I cook up these explicit Eilenberg-Zilber maps?