Projective spaces are usually defined as the quotient of a vector space (by the equivalence relation that identifies collinear vectors). However, in my opinion, projective spaces seem intuitively less structured and more primitive than vector spaces. Is there a way to reverse the order of definition?
Projective spaces can be defined axiomatically (albeit in a more geometric way than vector spaces). Can you extract the structure of a vector space from an underlying projective space?
The axioms of vector spaces usually talk about both the vector space itself, and the underlying scalar field, and the interaction between these two. So if you start off with a projective space, you'll need to extract a field from that first. I don't know about higher dimensions, but in the case of the plane, this is possible if and only if the theorem of Pappos holds in that projective plane. Which is not true in all projective planes. A key ingredient for the construction of a field from geometric facts are the constructions described by von Staudt which translate algebraic operations like addition and multiplication into geometric configurations. Once you have the field, it should be possible to work out the vector axioms for an arbitrarily chosen origin and a likewise arbitrarily chosen line at infinity. I don't have details on that.