The Galois number \(G_n^q\) is the total number of linear subspaces of \(\text{GF}(q)^n\). Is it possible to partition the lattice of subspaces of \(\text{GF}(q)^n\) into two intervals of length \(n-1\) and \(q^{n-1}-1\) intervals of length \(n-2\), for \(n\geq 2\)? The authors consider such interval decompositions for vector spaces \({\mathbb F}^n\) of finite dimension over arbitrary fields \({\mathbb F}\) and show that the existence of such a decomposition is equivalent to the existence of so called pointwise irreflexive and antisymmetric linear forms. This implies that for \(n\geq 3\) an interval decomposition of \({\mathbb F}^n\) exists only if \({\mathbb F}^{n-1}\) admits an interval decomposition. They show that \({\mathbb R}^n\) has an interval decomposition, while \(\text{GF}(2)^n\) has an interval decomposition if and only if \(n\leq 4\). They present an interval decomposition of \(\text{GF}(3)^5\).