A moment-angle complex $\mathcal{Z}_K$ is a compact topological space
associated with a finite simplicial complex $K$. It is realized as a subspace
of a polydisk $(D^2)^m$, where $m$ is the number of vertices in $K$ and $D^2$
is the unit disk of the complex numbers $\C$, and the natural action of a torus
$(S^1)^m$ on $(D^2)^m$ leaves $\mathcal{Z}_K$ invariant. The Buchstaber
invariant $s(K)$ of $K$ is the maximum integer for which there is a subtorus of
rank $s(K)$ acting on $\mathcal{Z}_K$ freely.

The story above goes over the real numbers $\R$ in place of $\C$ and a real
analogue of the Buchstaber invariant, denoted $s_\R(K)$, can be defined for $K$
and $s(K)\leqq s_\R(K)$. In this paper we will make some computations of
$s_\R(K)$ when $K$ is a skeleton of a simplex. We take two approaches to find
$s_\R(K)$ and the latter one turns out to be a problem of integer linear
programming and of independent interest.