We present an integer linear formulation that uses the so-called "distance variables" to solve the quadratic assignment problem (QAP). The model involves O(n^2 ) variables. Valid equalities and inequalities are additionally proposed. We further improved the model by using metric properties as well as an algebraic characterization of the Manhattan distance matrices that Mittelman and Peng recently proved for the special case of problems on grid graphs. We numerically tested the lower bound provided by the linear relaxation using instances of the quadratic assignment problem library (QAPLIB). Our results are compared with the best known lower bounds. For all instances, the formulation gives a very competitive lower bound in a short computational time, improving seven best lower bounds of QAPLIB instances for which no optimality proofs exist.