There is a completely explicit simplicial set realizing to $K(A,n)$ for each $n$, consisting of $\bar{A}[S^n]=\mathrm{ker}(A[S^n]\to A[*])$ (the kernel is taken levelwise, and $A[X]$ for each simplicial set $X$ is obtained by taking the free $A$-module levelwise). Here I let $S^n=\Delta^n/\partial \Delta^n$. I don't think you can get more explicit than that.

There is a completely explicit simplicial set realizing to $K(A,n)$ for each $n$, coming from the Dold-Kan correspondence.

It is defined as $$\bar{A}[S^n]=\mathrm{ker}(A[S^n]\to A[*])$$ (the kernel is taken levelwise, and $A[X]$ for each simplicial set $X$ is obtained by taking the free $A$-module levelwise). Here I let $S^n=\Delta^n/\partial \Delta^n$. 

I don't think you can get more explicit than that.