  We prove the invariance of homogeneous second-order Hamiltonian operators
under the action of projective reciprocal transformations. We establish a
correspondence between such operators in dimension $n$ and $3$-forms in
dimension $n + 1$. In this way we classify second order Hamiltonian operators
using the known classification of $3$-forms in dimensions $\leq$ 9. Systems of
first-order conservation laws that are Hamiltonian with respect to such
operators are also explicitly found. The integrability of the systems is
discussed in detail.
