This question is somewhat simple:
If we write the transposition of a matrix like this:
A_transposed = T.A
where T is the operator performing the transposition, can I find a matrix form for this operator, that is independent of the matrix A ? (In other words that would transpose any matrix the size of A?
No, you can't.
Take $A = I_n$, this would imply that $T = I_n$, but clearly, $A^T \neq I_n A$ if $A$ is not symetric