To answer this slightly evasively, the biggest responsibility of a manager is to handle any day-to-day situation that isn't on anyone's job description. If something isn't anyone else's problem, you make it your problem and figure out how to deal with it. Why is a project behind schedule? Why aren't these two engineers/teams/divisions talking to each other? Why is there an uptick of bugs in recent releases? How do you prepare your organization to respond to service outages or security breaches?

The ability to recognize and deal with such unknowns is the most useful skill for a manager.