Yes, and it has been done.
In 2010, the first human-powered ornithopter recorded its first flight.
It took a very light aircraft and a lot of power from the pilot, about 650 W, so the flight only lasted 20 seconds. It only sustained level flight, requiring assistance to take off.
To put this into perspective, a good power target for cyclists is 250 W. A top competitor will put out 300 W over a race and 400 W over 10-15 minutes, and over 1000 W for 10-20 seconds. So it's possible (not certain) that a top Tour de France cyclist could hold it in the air for a couple minutes, or even have the stamina for a takeoff. But it's still more power than any human can sustain long enough to cover a mile.
Meanwhile, the record for propeller-driven human powered aircraft, without flapping wings, currently holds at 115 km. This was achieved with autonomous launch.
The ornithopter in question isn't exactly an "exoskeleton", though, because an exoskeleton is not optimal for the job. Human arms are unsuitable for flying or sustaining high power output in general, so all successful human-powered aircraft derive power from the pilot's legs. Typically it's a cycling motion, but a straight leg press machine motion is used in this particular case.