Maybe you know something I don't, but I don't find these statements about the determinant and trace intuitive just from knowing that matrix multiplication corresponds to composition of linear maps. They can be made intuitive by appealing to geometry, but I don't see how to do that just by staring at the algebraic picture.
Actually, proving them without picking a basis and then showing that what you get is invariant under base change is quite nontrivial and involves throwing a lot of heavy machinery around; see for example Coffman's non-coordinate proof of the statement about the trace in http://users.ipfw.edu/CoffmanA/pdf/book.pdf
For the determinant, if you want to do everything without picking a basis, you're basically proving that the exterior power operation defines a functor on the category of finite-dimensional vector spaces, which isn't that bad to do, but you somehow have to explain what all those words mean along the way.