I have mounted a lot of 15x15 on 20x24, and it works okay. The next standard size up, 22x28, is nicer, but I find it a bit long for the width. For 16x20 I prefer 24x28 -- might be over-kill for 15x15, but then again, it might give the images a nice formal presentation with 4.5" sides, 5" (or 5.5") top and 8" (or 7.5") bottom.
Take a look again at how you would center on the mat -- consider having more space on top than on the sides. I think that works fine for square images on vertical rectangles.