  We consider the space $\mathcal M$ of ordered quadruples of distinct points
in the boundary of complex hyperbolic $n$-space, $\ch{n},$ up to its
holomorphic isometry group ${\rm PU}(n,1).$ One of the important problems in
complex hyperbolic geometry is to construct and describe a moduli space for
$\mathcal M$. For $n=2$, this problem was considered by Falbel, Parker, and
Platis. The main purpose of this paper is to construct a moduli space for
$\mathcal M $ for any dimension $n \geq 1$. The major innovation in our paper
is the use of the Gram matrix instead of a standard position of points.
