Equation for trace of a matrix is implied by the properties of trace???

[sto]

Let A,B be nxn matrices over a field F and define a function tr on the
nxn matrices by the relation tr(A) = sum_i a_ii.

It is simple to show that, for u,v in F,

tr( u A + v B ) = u tr(A) + v tr(B)

and that

tr(AB) = tr(BA)

in which case, if PP' = P'P = I and C = P' A P, then

tr(C) = tr(P' {A P})
= tr({AP} P')
= tr(AI)
= tr(A)

so that similar matrices have the same trace. Furthermore, it is clear that

tr(I) = sum_i (1) = n

However, if I instead define tr as a linear map on the nxn matrices
having the property that tr(AB) = tr(BA) and tr(I) = n, how does that
imply that the trace must satisfy the relation tr(A) = sum_i (a_ii)? In
other words, how do you prove that the trace is the only linear function
having these two properties?

I suspect this has something to do with the linearity of tr and the fact
that a linear function is fully specified by its action on a basis. But
I don't see how the relation tr(I) = n would be enough to solve the
problem that way. Recall also that not all matrices are similar to a
diagonal matrix, so I don't see how similarity can be useful here.

Thanks
-sto