examples of managerial skills @ a very young age

The best kind of managerial experience for b-school is when you have supervised and been responsible for the performance of others. Whether it be peers or subordinates, what b-schools look for is leadership as exhibited in this way. Beyond that, you can demonstrate various forms of project management or task management, but leadership of others is the most valuable experience.

I agree with Bryant (that Adcoms are interested in leadership). However, you also referred to social situations - while professional experiences are best, it can be appropriate to talk about "managing" something outside of work. A large fundraiser or social event might work (think about things that involve delegating, overseeing many tasks, resolving conflicts, motivating a team).

I have seen people write successful essays about running a household, serving as a camp counselor and overseeing sports teams as well. Just focus on the skills (professionalism, positive attitude, listening to others and getting "buy-in", training, mentoring, etc.) that make a good manager!