The Brieskorn and Orlik-Solomon theorems state that the cohomology ring of the complement of a complex hyperplane arrangement is isomorphic to the Orlik-Solomon algebra of its underlying matroid. In this talk, I will show that the homotopy sphere arrangements arising as homotopy colimits of diagrams of spaces on the geometric lattice of a matroid can be embedded into topological spheres when the codimension is greater than or equal to two. From this, we obtain a Goresky-MacPherson type formula for the cohomology groups of the complements of these arrangements and conjecture a cohomological interpretation for the Orlik-Solomon algebra of any matroid. This is joint work with Alex Engstrom.