For any reduced decomposition i = (i1, i2,... , iN) of a permutation w and any ring R we construct a bijection Pi : (x1,x2,... , xN) --> P_{i1}(x1) P_{i2}(x2) ... P_{iN}(xN) from RN to the Schubert cell of w, where P_{i1}(x1), P_{i2}(x2),... , P_{iN}(xN) stand for certain elementary matrices satisfying Coxeter-type relations. We show how to factor explicitly any element of a Schubert cell into a product of such matrices. We apply this to give a one-to-one correspondence between the reduced decompositions of w and the injective balanced labellings of the diagram of w, and to characterize commutation classes of reduced decompositions.