  In this paper, we consider a generalized version of the rectilinear crossing
number problem of drawing complete graphs on a plane. The minimum number of
crossing pairs of hyperedges in the $d$-dimensional rectilinear drawing of a
$d$-uniform hypergraph is known as the $d$-dimensional rectilinear crossing
number of the hypergraph. The currently best-known lower bound on the
$d$-dimensional rectilinear crossing number of a complete $d$-uniform
hypergraph with $n$ vertices in general position in $\mathbb{R}^d$ is
$\Omega(\frac{2^d}{\sqrt{d}} \log d) {n \choose 2d}$. In this paper, we improve
this lower bound to $\Omega(2^d) {n \choose 2d}$. We also consider the special
case when all the vertices of a $d$-uniform hypergraph are placed on the
$d$-dimensional moment curve. For such complete $d$-uniform hypergraphs with
$n$ vertices, we show that the number of pairwise crossing hyperedges is
$\Theta(\frac{4^d}{\sqrt{d}}) {n \choose 2d}$.
