Hmmm, opening the window at 30.000ft, where the air temperature is about -40°C/-40°F, air speed is extreme, you can't survive without oxygen, and you want to put your arm outside???
Well, if you really do, you'll get a sunburn quite fast. I got a sun burn within an hour in an altitude of 3500m / 12000ft, though I was tanned due to two weeks at the beach.
The ozone did already absorb a large fraction of the UV-C radiation, but as most of the air mass is already below you, the rest of the sun light will not have been filtered so much.
So, let's close the window.
To my knowledge, side windows and canopies are made of some kind of polycarbonate
like or Plexiglass.
CERN measured the transmission of light through a 2mm sheet of plexigas and came to the result that no light below ~350nm, passes this sheet. Depending on source, the UV region starts below 380-400nm.
Keep in mind that the windows are usually thicker, which makes this step in the diagram much steeper.
So, there isn't much UV left which can cause a sun burn, but let's take the data from rredc.nrel.gov to calculate how much light you get and compare it to the light at ground. I found a function describing the data from CERN quite well, and applied it to the sun spectrum data (without atmosphere) twice, to get a Plexiglass window of 4mm thickness. So, this is the worst-case, with no air above the aircraft pre-filtering the sunlight.
Here is the result, which shows that sunlight is much more intense, and its color is heavily shifted to blue. But the intensity drops to zero, just where the UV-A range starts. Even, if you put the beginning of the UV-A range to 400nm, the intensity is still lower than at the ground.
Wind shields of larger aircraft are usually a quite thick composite material of glass and some plastics, which should absorb UV the same way as polycarbonate does, if not better.
So, I don't see any evidence that you get more UV light in an aircraft than on ground.
However, you should not underestimate cosmic radiation, which is absorbed by our atmosphere quite well on the ground, but not at high altitude. A transatlantic flight is equal to an X-ray photography of your chest. Flying staff is an occupation group exposed to much higher radiation doses than any other group and it's dose usually is supervised.
As @PeterKämpf pointed out, Plexiglass is only one material used, and it is not a polycarbonate, which I didn't notice. But polycarbonates don't differ much concerning UV transmission, as the following diagram shows. However, I also read that polycarbonates - in contrast to plexiglass - are not UV hard and need an UV blocking coating when used outside.