They are both correct, but this doesn't work with all verbs. For example, you can say 'cooking skills', but not 'cook skills', the same as you can say 'cooking club' but not 'cook club'. The reason you can say both 'dancing skills' and 'dance skills' is because 'dance' is a noun which is the same as the art form - like 'music skills'.

Good morning! It is dancing skills. Dancing is an action word and dance is a noun.

I am not an English teacher, but the phrase dance club is used much more than dancing club. But, saying " improving my dance skills, and dancing skills" are both used regularly. Both are correct. Sorry I can not explain why they are correct compared to dance and dancing club, but they are. Its the English language thing...