As well as being used to make suggestions for actions, (으)ㄹ 까요? can be used to ask for opinions on doing something:
– 문을 닫을 까요?
– Shall I close the door?
– 네, 문을 닫으세요.
– Yes, please close the door.
– 차나 커피 드릴까요?
– Would you like some tea or coffee?
– 네, 정말 목이 말라요. 커피 한 장 주세요.
– Yes, I’m really thirsty. Please make me a cup of coffee.
무슨 영화를 볼까요?
Which movie would you like to watch?