No, pilots generally do not do that. At 22,000 feet you have time to get out the oxygen if the pressurization fails.
(2) Whenever a pressurized aircraft is operated at altitudes above 25,000 feet through 35,000 feet MSL, unless each pilot has an approved quick-donning type oxygen maskâ"
(i) At least one pilot at the controls shall wear, secured and sealed, an oxygen mask that either supplies oxygen at all times or automatically supplies oxygen whenever the cabin pressure altitude exceeds 12,000 feet MSL; and
(ii) During that flight, each other pilot on flight deck duty shall have an oxygen mask, connected to an oxygen supply, located so as to allow immediate placing of the mask on the pilot's face sealed and secured for use.
Every now and again there is an issue where the plane does not ever pressurize in the first place and no one notices. Not every airplane has a cabin pressure alarm.