What are the advantages of Skip-Gram methods? Concerning the notion of word embeddings, Skip-Gram methods aim for computing the probability of a word given its neighborhood. I do not understand the rationale behind it, since it is possible to infer this information by looking directly into the co-occurrence matrix.

In general, I cannot understand those methods aiming to capture as much relevant information from the original co-occurrence matrix as possible. Isn't it easier to work on the co-occurrence matrix directly?

Thank you very much in advance.
Basically, skip-gram aim for computing the probability of a context given a word (it is CBOW that does the opposite). BUT, it does not simply learn a co-occurrence matrix, it compresses the information in a low dimensional space (for example 300 for a 100000 original dimension). By doing this, it learns continuous low dimensional representation for the words.

It is proved that word2vec actually factorizes a word-context pointwise mutual information matrix (close to the co-occurrence matrix). I think this article might give you a better understanding of the underlying processes.