It is possible. The shadow of the moon goes too fast to keep up with that, but there are other reasons to do it. Firstly, when you fly over the clouds, you can be sure they don't obstruct your view. Second reason is that sometimes solar eclipse, or the best part of it, only appear over an ocean.
Here is an article from last year. Couple of interesting quotes from it:
"Taking to the skies during the sun being eclipsed out of view behind the moon, is not a new concept. In fact in 1887, Dmitri Mendeleev, the Russian inventor, was so keen to see the eclipse without cloud obstruction that he rode in a hot air balloon when it took place."
"In the past private operators have capitalised on the event by offering special eclipse flights."