  The rigged configuration realization $RC(\infty)$ of the crystal $B(\infty)$
was originally presented as a certain connected component within a larger
crystal. In this work, we make the realization more concrete by identifying the
elements of $RC(\infty)$ explicitly for the $A_n$-type case. Two separate
descriptions of $RC(\infty)$ are obtained. These lead naturally to isomorphisms
$RC(\infty)\cong T(\infty)$ and $RC(\infty)\cong \bar{T}(\infty)$, i.e., those
with the marginally large tableau and marginally large reverse tableau
realizations of $B(\infty)$, that may be computed explicitly. We also present
two descriptions of the irreducible highest weight crystal $B(\lambda)$ in
terms of rigged configurations. These are obtained by combining our two
descriptions of $RC(\infty)$, the two mentioned isomorphisms, and two existing
realizations of $B(\lambda)$ that were based on $T(\infty)$ and
$\bar{T}(\infty)$.
