such that for any vector $\mathbf{v},$ $\mathbf{R} \mathbf{v}$ is the reflection of $\mathbf{v}$ through the $xy$-plane.

Reflection  of a vector across the xy plane  just reverses the vector components

The reflection matrix, R,  is

[  0    1

   1    0 ]

Proof

⟨a,b⟩   [  0  1      =     ⟨ a* 0  + b * 1,  a*1  + b*0 ⟩  =  ⟨b,a⟩

            1   0 ]