The hole in the middle of skate bearings is 8mm. An 8mm threaded rod will fit it perfectly.
Converting to Imperial measurements, 8mm is .3149" which is very close to .3125". 5/16" threaded rod. No, it's not a *perfect* fit, but it's close enough. And, you can shim it to perfect size by filling in the gap with something like wire, if desired.
-- Chuck Knight