In this paper discrete quasi-copulas (defined on a square grid I2 n of [0,1]) are studied and it is proved that they can be represented by means of a special class of matrices with entries in [-1,1]. Special considerations are made for the case of irreducible discrete quasi-copulas (those with range In) defined on the finite chain In, showing that they can be represented through Alternating-Sign Matrices and that they generate all discrete quasi-copulas through convex sums.