"I want to go to Korea."
Translation:저는 한국에 가고 싶어요.
That is true. But to be 100% grammatically correct we should add it in. It's similar in English to questions like "Going out?" when directed at someone leaving the house. In context it makes sense. But the true question is "are you going out?" and is implied by context. It all makes sense. But in a classroom, as an ESL teacher I wouldn't accept "going out?" as an answer on a test. Maybe partial credit at best. In a casual environment, chatting in class, its acceptable then. But not in a test. But that's just my opinion.