  A Formal Orthogonal Pair is a pair $(A,B)$ of symbolic rectangular matrices
such that $AB^T=0$. It can be applied for the construction of Hadamard and
Weighing matrices. In this paper we introduce a systematic way for constructing
such pairs. Our method involves Representation Theory and Group Cohomology. The
orthogonality property is a consequence of non-vanishing maps between certain
cohomology groups. This construction has strong connections to the theory of
Association Schemes and (weighted) Coherent Configurations. Our techniques are
also capable for producing (anti-) amicable pairs. A handful of examples are
given.
