Not usually, but you're right that it's not straightforward. The lone pair can usually flip back and forth between the other three substituents, like turning a sock inside out. Sometimes though, if a molecule is rigid enough, this isn't possible, and it can form a chiral center. I'm willing to bet we won't be asked about something like that, but you never know.
Dasypus is right, nitrogen is essentially never chiral because the lone pair exists in an extremely fast equilibrium between the two possible positions.