It's not that it's "staying up there", it's that it never
has a chance to fall. Ozone is FORMED up there, and it
breaks apart again before it has a chance to fall.
Ozone is formed when UV radiation hits oxygen (O2),
splitting it apart. Oxygen atoms hate to be alone, so they
recombine with an O2 molecule to make O3, ozone. Before the
ozone has a chance to fall, IT gets hit with MORE uv
radiation, turning it back into oxygen. Ozone just keeps
getting made and destroyed before it has a chance to fall.