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How is this?

sage: dim=4                                
sage: F = PolynomialRing(QQ, dim,'X')      
sage: I = F.ideal([x*y for x,y in tuples(F.gens(),2)])
sage: pol = [I.reduce(F.random_element()) for i in range(dim)]
sage: pol
[-3*X2, -39/2*X1 - 1/18*X2, -1/10*X0 - 3*X1 - X2 - 1/5*X3, X2 + 6*X3]
sage: matrix(dim,lambda i,j:pol[i].coefficient(F.gen(j)))
[    0     0    -3     0]
[    0 -39/2 -1/18     0]
[-1/10    -3    -1  -1/5]
[    0     0     1     6]