  We investigate the combinatorial structure of subspaces of the exterior
algebra of a finite-dimensional real vector space, working in parallel with the
extremal combinatorics of hypergraphs. Using initial monomials, projections of
the underlying vector space onto subspaces, and the interior product, we find
analogues of local and global LYM inequalities, the Erd\H{o}s-Ko-Rado theorem,
and the Ahlswede-Khachatrian bound for $t$-intersecting hypergraphs.
  Using these tools, we prove a new extension of the Two Families Theorem of
Bollob\'{a}s, giving a weighted bound for subspace configurations satisfying a
skew cross-intersection condition. We also verify a recent conjecture of
Gerbner, Keszegh, Methuku, Abhishek, Nagy, Patk\'{o}s, Tompkins, and Xiao on
pairs of set systems satisfying both an intersection and a cross-intersection
condition.
