The key to the answer is what kind of oil do you have, there are differences. In general, oils with big amount of double bonds MAY oxidize by oxygen from air. The process is radical, UV-induced, and very slow. Some other oils, with no or too little double bonds do not oxidize. and remain mostly same.
The process of oxidation (drying) initially produces hydroperoxides, later able to serve as a source of more free radicals, leading eventually to inter-molecular linkages. I found contradicting notions for what these linkages consist of (it is either oxygen bridges or direct links similar to polymers), but the end process if well known and used in varnishes. However, varnishes has extra components: they are usually diluted by some solvent to reduce viscosity and always contain a metal catalyst to accelerate drying of the oil, that also is usually heated for prolonged period of time, again, to increase the speed of drying.
The considerations above ignore obvious process of evaporation of light impurities and possibility of bio-degradation (I'm not aware of bio-degradation processes for pure fatty oils, but there are bacteria that live in almost undiluted toluene and there are bacteria that thrive on petrol, so I'm open to this idea).