Consider the matrices with a single 1 on the diagonal, 0 elsewhere. There are of these. There are also matrices with two 1s in symmetric positions off diagonal and with a 1 and -1 in symmetric positions off diagonal (so the resultant matrix is antisymmetric).

These form a basis for (they are clearly linearly independent, and there are of them). The symmetric matrices we have defined has eigenvalue 1 under , and the antisymmetric matrices we have defined has eigenvalue -1. It should be obvious what the matrix of looks like with respect to this basis: 1s and -1s down the diagonal, zero elsewhere.

The determinant of is the product of eigenvalues: