Increase spacing of every third row in Grid

I have a grid whose number of rows is a multiple of 3, for example

Grid[{{a, a},{a, a},{a, a},{a, a},{a, a},{a, a}}]


Is there an automatic way using the option spacings, so that at every third row the spacing is increased? I tried

Grid[{{a, a},{a, a},{a, a},{a, a},{a, a},{a, a}},Spacings -> {0, {0, 0, 3}}]


but this only inserts a spacing of 3 in the third row without repeating the pattern.

Grid[{{a, a}, {a, a}, {a, a}, {a, a}, {a, a}, {a, a}},