  We enhance the quandle counting invariants of oriented classical and virtual
knots and links using a construction similar to quandle modules but inspired by
symplectic quandle operations rather than Alexander quandle operations. Given a
finite quandle $X$ and a vector space $V$ over a field, sets of bilinear forms
on $V$ indexed by pairs of elements of $X$ satisfying certain conditions yield
new enhanced multiset- and polynomial-valued invariants of oriented classical
and virtual knots and links. We provide examples to illustrate the computation
of the invariants and to show that the enhancement is proper.
