To be self-aware is, in a sense, to understand that you exist as a separate entity from the rest of the world. One creative way of assessing self-awareness is through a self-recognition test.
From this video, you should be able to define self-recognition and how we can test for a sense of self-awareness it with children and chimpanzees. Be sure to note the age at which self-recognition develops in children.
Watch: Self-Recognition in Apes (http://youtu.be/vJFo3trMuD8?t=1m14s)
Watch: Is Elephant Self Aware and Conscious? (https://youtu.be/0_qie0HRTdQ)
(note - don't worry about the stages, just watch to see another good example)