Not an answer, but an amusing observation: The determinant of the matrix usually (but not alwasy) has denominator a perfect square (empirically), while the numerator always seems to have a huge prime factor.
n=1: det = 1/1
n=2: det=7/(12^2)
n=3 det = 647/(2160^2)
n=4 det = (19 * 571)/672000
n=5 det = (179 * 179357)/(7*4233600000^2)
n=6 det = (97 * 157 * 384191938531)/(186313420339200000^2)
n=7 det = (23 * 1280587616051046200369)/(2067909047925770649600000^2)
n=8 det = (317 * 6337 * 25997 * 87403 * 511645991608091)/(365356847125734485878112256000000^2)