do i need a skimmer on my nano with a refugium

okay i know this is a commonly asked question but its a little more complicated than the title itself. i have a 10 gallon nano tank but im adding another 10 gallon sump/refugium to my tank. i know it might not seem like a lot but i figured any sump is better than no sump, and space is limiting the size. im just wondering if i need a protein skimmer if i already have a refugium, since they both do the same thing and skimmers are not usually put in nano tanks anyways. im very religious about water changes, i alternate between 10% and 20% weekly. thanks for the advice