In 2D, the rotation matrix is 2x2, R(theta), in 3D you have 3 3x3 matrices, Rx(theta), Ry(theta) and Rz(theta), as you can see here:

https://en.wikipedia.org/wiki/Rotation_matrix

In n dimension, you have n rotation matrices, all of them nxn?If so, how do you construct them?

Do they look like this in 4D?

https://ksgamedev.files.wordpress.com/2010/01/matrix-rotation2.png