Is there a matrix that can be used to find the transpose of a matrix?

There is no general transposer matrix.. For this, e.g. note that the only transposer for the identity in any dimension is the identity as the equation

$$T^II=I$$

needs to be fulfilled, but $T^II=T^I$.

However, for e.g. for non-symmetric matrices $A$, the identity is definitely not the transposer, as $A^\top\neq A$, but $IA=A$.

Now, we may try to construct the transposer matrix by considering $X=(x_{ij})_{i,j\leq n}$ and $A=(a_{ij})_{i,j\leq n}$, and then solving the equation $XA=A^\top$ for $X$. I.e. by the laws of matrix multiplication, you have to have

$$a_{ij}=\sum_{k=1}^n a_{jk}x_{ik}$$

for all $i,j$. However, this system of linear equations is not always solvable (only if $A$ is not symmetric). For this, consider the following example:

Look at $A=\begin{pmatrix}0 &1\\0 &0\end{pmatrix}$, then

$$\begin{pmatrix}x &y\\z &w\end{pmatrix}\begin{pmatrix}0 &1\\0 &0\end{pmatrix}=\begin{pmatrix}0 &x\\0 &z\end{pmatrix}\neq\begin{pmatrix}0 &0\\1 &0\end{pmatrix}$$