Radiator hose change. Leave cap off to remove air after replacing?

First time replacing the radiator hose as I’m getting antifreeze spray from the hose area and thought I’d go cheap initially. I’m also replacing the hose to the overflow box as it’s wet and I have antifreeze spray around the cap and on the actual box. Do I leave the radiator cap off after install to remove the air? 2000 3.4L. The truck is not overheating but the overflow is draining quick. The fluid stays full in the radiator.

The radiator is about 13 years old and I haven’t seen any leaks so my thought is if I replace the hose, clean up the spray, if it’s still blowing antifreeze, then I’ll have the radiator tested.