Skeleton of a category

Would you let me know when the category has an equivalent skeleton? (The definition of the skeleton subcategory that I have in mind is from MacLane p91: a full subcategory such that for any object in the original category, there exists a unique isomorphic object in the skeleton subcategory.) My question is mainly about when I can use the choice axiom without causing contradiction. For instance, I heard that the category of abelian groups doesn't have an equivalent skeleton subcategory. Thank you very much, Hongseok