I'm good at dancing is the correct one. But it's better to say "I'm a good dancer".

ok, to understand this you have to understand the use of nouns and verbs. Chess is a noun and dance is verb. You have to only use "dancing" because if you use "dance", the meaning changes. In the sentence, 'im good at dance' you no longer mean that you are good at dancing but good at 'dance', in which the said 'dance' assumes a noun form and there for referring to something other than actual dancing.

Hi Avicenna! I had this doubt a time ago. You could search someting like " gerund after preposition". It has a rule . :)

Dancing is a noun (dancing is good for you), dance is a noun with a different meaning. I'm good at dancing. Painting is a noun (painting is relaxing), paint is a noun with a different meaning. I'm good at painting. Swimming is a noun (swimming is good exercise), swim is a noun with a different meaning. I'm good at swimming. Chess is a noun, there's no 'ing' form of the noun. I'm good at chess. Tennis is a noun, there's no 'ing' form of the noun. I'm good at tennis. If there is an 'ing' form of the noun, then that is the one we use after I'm good at. At least that's what I think.

One is a noun (dance), one is a verb (dancing). I think either is grammatically correct, but "I'm good at dancing" would be more natural and sound better.