Yes, there are such things, but they aren't really related to word2vec.
First of all, Bag of Words is the old "something2vec" approach of representing documents in a vector space. Of course, the resulting vectors are very sparse, and distance does not always behave well in such cases, and it is better to use dot-product based similarities like cosine.
But you can reduce the dimensionality of this vector space with SVD or NMF (Non-Negative Matrix Factorization), and this will give you what you want: dense vectors of small dimensionality for which you can compute the distances.
Then you can go further and use Topic Modelling, which also will give you a vectorial representation of documents, and you can use it for comparing them.