My wife and I spent about 36 hours in Madrid last summer - we were staying in Salamanca and popped through for the weekend.
On the Friday evening we went to a bull fight at the Plaza del Torros which was quite an experience although I wouldn't go if you're a fluffy-bunny-loving vegan.
On the Saturday we split up for a few hours (something we rarely do on our travels as our tastes overlap massively) and fell into our gender stereotype roles.
I went for an impressive guided tour of Real Madrid's Bernabeu stadium and she went shopping. London (where we live) has football grounds and shops but we were both impressed with our finds!
Other than that we had a nice walk through the park near the palace - due to the nice weather we chose that option rather than going round a gallery.
Enjoy.