I have never understood the trace map,not even after reading http://mathoverflow.net/questions/13526/geometric-interpretation-of-trace. The problem with many answers in the above discussion is the geometric intuition does not apply to other field.
As I don't want this to be closed, let me make the question more precise. Is there a definition of the trace map which
1) is basis independent, (there was a definition given by Sridhar Ramesh in the old post).
2) explains in an intuitive way why if $L$ is a finite separable extension of $K$, the map $ (x,y) \mapsto Tr(xy) $, where $x,y$ are in $L$, is a non degenerated bilinear form on L?