  Unordered flag manifolds are the manifolds of unordered $n$-tuple of mutually
orthogonal lines in $\mathbb{R}^n$. In this paper, we develop some basic tools
to compute the mod-$2$ cohomology groups of these spaces, and apply them for
explicit computation for small $n$. We show that this computation improves the
known estimate of the number of Auerbach bases of normed linear spaces of small
dimensions.
