Given a collection of $t$ subspaces in an $n$-dimensional vector space $W$ we can associate to them $t$ linear ideals in the symmetric algebra $\mathcal{𝒮}(W^{\ast })$. Conca and Herzog showed that the Castelnuovo–Mumford regularity of the product of $t$ linear ideals is equal to $t$. Derksen and Sidman showed that the Castelnuovo–Mumford regularity of the intersection of $t$ linear ideals is at most $t$. We show that analogous results hold when we work over the exterior algebra $\bigwedge (W^{\ast })$ (over a field of characteristic 0). To prove these results we rely on the functoriality of equivariant free resolutions and construct a functor $\mathrm{\Omega }$ from the category of polynomial functors to itself. The functor $\mathrm{\Omega }$ transforms resolutions of polynomial functors associated to subspace arrangements over the symmetric algebra to resolutions over the exterior algebra.