Because the plane has to stay over the river to avoid noise to the neighborhoods near the airport, when the river turns sharply north of the airport, so must the airplane, so your view of downtown (taking off looking out the right side) is VERY brief (5-10 seconds), and if you're not in the window seat you won't see anything at all.

I used to live directly under the flight path north of the airport and it's a much better view landing from the north than taking off to the north.

If you want the best view of downtown, visit the Wash Monument and view at your leisure. Enjoy DC