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For sure, I would spend 1 day in the city to walk the plazas, see the Palace (don't miss it!), and any other attractions that appeal to you. We saw the botanical gardens (weren't impressed), Reina Sofia (loved it), Prado (lovely museum, but not my preference of art), Retiro Park (great for strolling and people watching), Templo de Debod (kinda cool). Those, along with walks through Plaza Mayor & Puerto de Sol and some great leisurely meals, took us 2 full days to see. There are plenty of other things to see in Madrid as well to fill your third day.
If you'd rather, you can leave the city for a day trip for your 3rd day. You can see El Escorial, Toledo, Segovia, or Avila easily by train from Madrid (or guided tour, which is more $$ but more convenient). We chose Segovia and thought it was a great little town with a lot of personality.