What scenes in the drama best illustrate Nash's underdeveloped social skills?
I think that the start of the film is where scenes that best illustrate Nash's underdeveloped social skills are most evident. The opening sequence at the luncheon in which Nash is moving the glass around to create patterns of light reflected from the sun, as opposed to socializing with the other students would be one such moment. The obvious social tension between he and Martin Hansen would be another such instance. Hansen is much more socially savvy and personable than Nash could ever be. The fact that Hansen is able to manipulate this into making Nash even more socially awkward can be seen in their matches of Go and the interactions at the bar with women.
Nash is also seen as socially awkward in his first interaction with Alicia. In this sequence, Nash is late to teach a class. It becomes clear that Nash discredits teaching in favor of research and theoretical development. Nash runs to class in only an undershirt and slacks. After an extended tardy, he comes to class and begins teaching without any introduction or explanation to him being late. When the heat of the classroom is further stifled by an open window that Nash closes without any regard for his students, Alicia is the one who opens it and helps to illuminate his own awkwardness, something that causes him to take note of her and the fact that perhaps a more socially acceptable manner is something that he can adopt.