  We extend the notion of representation of a matroid to algebraic structures
that we call skew partial fields. Our definition of such representations
extends Tutte's definition, using chain groups. We show how such
representations behave under duality and minors, we extend Tutte's
representability criterion to this new class, and we study the generator
matrices of the chain groups. An example shows that the class of matroids
representable over a skew partial field properly contains the class of matroids
representable over a skew field.
  Next, we show that every multilinear representation of a matroid can be seen
as a representation over a skew partial field.
  Finally we study a class of matroids called quaternionic unimodular. We prove
a generalization of the Matrix Tree theorem for this class.
