Hey,

SK will be able to check your luggage through to your destination, but you probably have to check in for a boarding pass at the service centre at your connecting airport. Make sure you have at least 2 hours connection time, a little delay, you may miss your connection and SK may force you to buy a ticket.... I dont know how strict SQ are.

Like many have said before, 2 separate tickets may be trouble. I recommend you to pay a little extra to get on the same itinerary, you will then have no responsibility if you miss a connection