  The Orlik-Solomon algebra A of a matroid is isomorphic to the quotient of an
exterior algebra E by a defining ideal I. We find an explicit presentation of
the annihilator ideal of I or, equivalently, the E-module dual to A. As an
application of that we provide a necessary, combinatorial condition for the
algebra A to be quadratic. We show that this is stronger than matroid being
line-closed thereby resolving (negatively) a conjecture by Falk. We also show
that our condition is not sufficient for the quadraticity.
