  For a real affine hyperplane arrangement, we define an integer intersection
matrix with a natural $q$-deformation related to the intersections of bounded
chambers of the arrangement. By connecting the integer matrix to a bilinear
form of Schechtman-Varchenko, we show that there is a closed formula for its
determinant that only depends on the combinatorics of the underlying matroid.
We conjecture an analogous formula for its $q$-deformation. Our work also
applies more generally in the setting of affine oriented matroids.
  Additionally, we give a representation-theoretic interpretation of our
$q$-intersection matrix using Braden-Licata-Proudfoot-Websters's hypertoric
category $\mathcal{O}$ (or more generally Kowalenko-Mautner's category
$\mathcal{O}$ for oriented matroid programs). This paper is part of a broader
program to categorify matroidal Schur algebras defined by Braden-Mautner.
