What kind of skills are really required from high-scale manager?

It happens very frequently, you are looking for a managerial position and once again you found that the experience in management teams with dozens of developers is truly required. Is it really required? If you asked me - my answer will possible wonder you: in a normal organization there is no manager that is actively managing more then 10-15 people. How could it be? It is not so difficult, just think about that: CEO is managing a team of VPs and COs, totally no more then 15; every one of VP's or CO's is managing a team of division managers, totally no more then 15; and so force. So, from the perspective of number of people that is under the direct responsibility of CEO is more or less equal to a number of developers that team leader is responsible on. So, what is really made a difference? This is exactly what is required for such positions: the ability to make decisions (sometimes those decisions are wrong at the long run, however when it is done it is done), the vision, the ability to define strict rules, etc.