Yes you can, but there is a couple of caveats. The first is that the new washer hookup will need to be 2 inch pipe, instead of 1-1/2 inch. The reason is, the sink acted like a storage tank for the water pumped from your washer. It allowed the water to slowly flow out of the drain at the rate that the drain could absorb. If you try to use an 1-1/2 trap and drain for the new standpipe for the washer, it will overflow, because the washer will pump more water into the drain than can flow down the pipe. A washer uses anywhere from 15 gallons of water on the energy saver cycle to 30 gallons on the large load cycle. In addition it pumps it out very fast, typically at a much greater rate than the pipe is rated for.
The second is you may have to go down into the floor or to the main stack to find 2 inch pipe. Which may, as LJ says require some drywall and flooring work. Where you could possibly install a sink all by yourself, running all the piping necessary to retrofit for a washer will require a plumber.