I understand that $(\mathbb{Z}^n)^* = \operatorname{Hom}(\mathbb{Z}^n, \mathbb{Z})$. But is there a generic description or more intuitive isomorphism for that space? I'm trying to use Hungerford as a reference, but it doesn't seem to help with trying to calculate what the dual space is.
This is stemming from looking at something in homological algebra. So we have a chain complex. $C_n\rightarrow \cdots \rightarrow C_0$ and each $C_i$ is a free abelian group (or $\mathbb Z$-module) with $n_i$ generators. We $\hom(-,\mathbb{Z})$ everything and end up with $$\hom(C_0,\mathbb{Z}) \rightarrow \cdots \rightarrow \hom(C_n, \mathbb{Z}).$$ I'm interested in understanding the derived functor $\operatorname{Ext}^i$ and it seems like understanding the hom-group is the first step to that.