Constructing Basis Mirroring Matrix

Question

I have a $2^N$ dimensional basis of a vectorspace $$ |s_1 \rangle \otimes |s_2 \rangle \otimes ... \otimes |s_N \rangle $$ where $s_i \in \{\uparrow,\downarrow \}$ is the standard basis. I need to construct $2^N \times 2^N$ a matrix $M$ that mirrors my kronecker product of basis vectors along the middle $$ M |s_1 \rangle \otimes |s_2 \rangle \otimes ... \otimes |s_N \rangle = |s_N \rangle \otimes |s_{N-1} \rangle \otimes ... \otimes |s_1 \rangle. $$ I have calculated it for $N=1,2,3$ by hand $$ M(N=1)=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$ $$ M(N=2)=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$ $$ M(N=3)=\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} $$ I would like to have a general formula for any $N$. It might be possible to construct it in terms of commutation matrices. Also does this Matrix have a special name?