You have to change permissions. It's a little problem though, fat32 doesn't support permissions, so you have to chown and chmod the whole partition. Say that your partition is /mnt/fat
Then you do a chown user /mnt/fat if you want user to own it.
Then you do a chown 755 /mnt/fat i.e. because then user can do everything with the stuff in it and everybody else can read and execute everything in it.
If it complains, umount it first: #umount /mnt/fat
and try again.