Smithsonian Institution National Museum of Natural History
The Smithsonian Institution's National Museum of Natural History is comparable in size to New York's American Museum of Natural History, but less of its floor space is devoted to dinosaurs. Even so, you'll find a considerable number of dinosaur skeletons here--not fabricated reproductions, but actual fossils, including the famous "roadkill" Triceratops that (until the 1990's) was the most complete in the world, the tyrannosaur Gorgosaurus, and the sauropod Diplodocus. Most of these reconstructions can be viewed in the exhibit "The Last American Dinosaurs: Discovering a Lost World," along with lesser-known genera like Thescelosaurus and Sphaerotholus.
One of the oldest dinosaur museums in the world, the National Museum of Natural History has had to periodically remove its exhibits from display in order to restore or refurbish them (or, in some cases, to completely repose them according to the latest theories of dinosaur physiology). For example, the Triceratops mentioned above has been given a complete facelift, as has the museum's famous Stegosaurus (which has been reoriented so that it appears to be reacting to the Allosaurus skeleton directly behind it, which clearly intends to eat it for lunch).
If you're interested in any fossils over and above dinosaurs, you'll unfortunately have to wait until 2019, as the National Museum prepares the National Fossil Hall for the public. If you simply can't wait, though, you can access a live view of the hall-in-progress at the museum's website.