Is there any way to see which Pokestops I have already spun?

There is a history log of everything you have done so far in terms of pokemon encounters, gym fights and Pokestops.

Select your trainer image in the bottom left of the main screen, then click on the "Journal" button just below your XP bar.

While this does not tell you the location of the pokestop, it does tell you when you went there and what you received.

Considering Niantic's previous game (Ingress), and how it tracked "unique portals" (their equivalent of pokestops/gyms), I would imagine the data is stored somewhere in their servers, but it is not made public. However, it could become public at some point in the future, and I would probably expect it to be connected to the journal (or just become a new feature).

There will be a white circle around the top of any previously unvisited stops!

Pokestops that you have not visited yet will have a white circle around them. That circle goes away after you spin them once.

Unvisited

Visited