As mentioned, ready-made square frames are not very common but whole frame supply companies for do-it-yourself framers like my company have no problem providing square frames or heavily bottom weighted mats such as in the examples above. In general, larger mat borders tend to give the image more of a gallery feel. For a square image 11x11 image, equal 4.5 inch mat borders resulting in a 20x20 inch frame would have nice gallery feel. Off the shelf frame sides are commonly available in 20 inch segments although still need a custom mat.
The heavy bottom weighted mat, where the bottom border is significantly larger than the top and side borders will also offer a nice gallery flare and in fact Nielsen actually makes a couple ready-made frames for 8x10 images in 16x20 frames. You can see pics here, and I have included the mat border dimensions at the bottom of the text on these two pages:
Nielsen Gallery Collection 8x10 (16x20) GF1950E
Nielsen Gallery Collection 8x10 (16x20) GF1950D
Nielsen use to carry an 11x14 frame with 5x5 opening but they discontinued it. Bottom weighted mats are almost never offered in ready-made frames because as you notice with the 8x10 you have to manufacture and stock two different sizes.
If you are going to bottom weight the top and side borders should be equal, or the top border should be within an inch of the side borders. If the frame is 11x14 or smaller I would suggest the top border be less than 1 inch larger than the side borders if not equal.