If you have a directory called Frame, and you want to change all the Frame and every thing under it, then do this
sudo chown -R osmc Frame
That will change the Frame owner to osmc, and every file and directory below that to osmc also.
(Sorry I forgot to add the sudo first time.)
If you are plugging you SD card into a Linux Mint system, check what your user id is on Mint:
brian@cyberman ~ $ id
Notice that my uid is 1000, which is the same uid as osmc:
uid=1000(osmc) gid=1000(osmc) groups=1000(osmc),4(adm),6(disk),7(lp),20(dialout),24(cdrom),29(audio),44(video)
In that case, you don’t need to change the owner, because it’s the same (1000).
If the user on Mint isn’t 1000, you could just use that user on Mint (by default the first user you created when installing Mint is 1000) Using a user with the same uid saves you the hassle of changing owners everytime you move the SD card.