I think you can look at the initial boot screen of Ubuntu MATE to determine if it is booting in UEFI or Legacy mode. If you see a "themed" (grey, transparent) screen with selections:
Try Ubuntu MATE without installing
Install Ubuntu MATE
Check memory
etc
you are booting in Legacy mode. If you see the same screen without theming (black background, white text) you are booting in UEFI mode.
I will try to find a link that shows the difference.
Found it. This site shows the difference in Ubuntu.
https://help.ubuntu.com/community/UEFI
With Legacy the boot screen is themed. With UEFI the boot screen has no theming. I think this applies to Ubuntu MATE as well, but I am not 100 % sure.