Cochains as a set of simplexes

Question

I know that if $X$ is a simplicial complex then $C_{n}$ is the free abelian group on the n-dim simplices (ex. $C_{0}$ is the free abelian group on vertices etc). Then, the elements of $C_{n}$ are called n-dim chains. Now, in order to get the cochains we replace the groups $C_{n}$ by $C^{*}_{n}=Hom(C_{n}, G)$, where $G$ is an abelian group. I am now reading a paper and it states that we think of an n-cochain as a set of n-simplexes. I don't understand how we can do this since the elements of $C_{n}$ are maps. Any help?