I have never understood the trace map,not even after reading 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?