How do I calculate the determinant of a 6x6 matrix in C++ really fast?

How do I calculate the determinant of a 6x6 matrix in C++ really fast?

In C++ I need to calculate the determinant of a 6x6 matrix really fast.
This is how I would do this for a 2x2 matrix:
double det2(double A[2][2]) {
return A[0][0]*A[1][1] - A[0][1]*A[1][0];
}
I want a similar function for the determinant of a 6x6 matrix but I do not
want to write it by hand since it contains 6! = 720 terms where each term
is the product of 6 elements in the matrix.
Therefore I want to use Leibniz formula:
static int perms6[720][6];
static int signs6[720];
double det6(double A[6][6]) {
double sum = 0.0;
for(int i = 0; i < 720; i++) {
int j0 = perms6[i][0];
int j1 = perms6[i][1];
int j2 = perms6[i][2];
int j3 = perms6[i][3];
int j4 = perms6[i][4];
int j5 = perms6[i][5];
sum +=
signs6[i]*A[0]*A[j0]*A[1]*A[j1]*A[2]*A[j2]*A[3]*A[j3]*A[4]*A[j4]*A[5]*A[j5];
}
return sum;
}
How do I find the permutations and the signs?
Is there some way I could get the compiler to do more of the work (e.g. C
macros or template metaprogramming) so that the function would be even
faster?