Veterans Saluting the Flag at Events - Sincere Question

I understand that, as honorably discharged veterans, we get the privilege of saluting the flag during the National Anthem. My question is this. We're all expected to remove our hats during the National Anthem, but we were taught to not salute without a cover on. So I've always been a little confused by that. Do we leave our hats/covers on so that we can salute?