n=5
K = QQ[flatten apply(n, i -> apply(n, j -> a_(i,j))),Degrees=> apply(n^2, i -> {0}),MonomialOrder=>Lex];
R=K[join(apply(n, i -> f_i),apply(n, j -> g_j)),MonomialOrder=>Lex];
M = matrix apply(n, i -> apply(n, j -> a_(i,j)));
I = matrix apply(n, i -> apply(n, j -> if i==j then 1_K else 0));
X1 = map(R,R,M++I);
X2 = map(R,R,I++(-transpose(M)));
F = sum apply(n, i -> f_i*g_i);
X1(F) + X2(F)==0

-- true for n=4, 5