In three dimensions, every rotation is in fact a rotation about some single axis. To specify a rotation in three dimensions, it is therefore necessary to specify this axis, and the angle about this axis through which to rotate. It takes two parameters, such as latitude and longitude, to determine the location of the axis, and a third to give the angle of rotation. These three parameters are collectively known as Euler angles.

Another way to describe rotations in three dimensions is to explicitly construct rotations (only) in the coordinate planes, and then argue that any rotation can be obtained by suitably combining these rotations. In particular, it is well-known that any rotation matrix in three dimensions can be written (in several ways) as the product of three such rotations in coordinate planes; this is in fact one way to construct the Euler angle representation to begin with.

Since a description in terms of generalized Euler angles is not available in higher dimensions, we focus instead on the alternative description in terms of rotations in coordinate planes.

Rotation matrices of the form given in the previous section generalize in an obvious way to higher dimensions: Just make the matrix bigger, and add $1$ and $0$ appropriately in the remaining entries. This construction leads to the three matrices \begin{align} R_x &= R_x(\alpha) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & -\sin\alpha \\ 0 & \sin\alpha & \cos\alpha \end{pmatrix}\\ R_y &= R_y(\alpha) = \begin{pmatrix} \sin\alpha & 0 & \cos\alpha \\ 0 & 1 & 0 \\ \cos\alpha & 0 & -\sin\alpha \end{pmatrix}\\ R_z &= R_z(\alpha) = \begin{pmatrix} \cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1 \end{pmatrix} \end{align} corresponding to rotations in the $yz$, $zx$, and $xy$ planes, respectively.

Thus, the orthogonal group in three dimensions is defined by \begin{equation} \SO(3) = \{ M\in\RR^{3\times 3} : M^T M = I, \det M = 1 \} \end{equation} but is generated by the matrices $R_x$, $R_y$, $R_z$. We write this relationship as either of \begin{equation} \SO(3) = \langle \{ R_x,R_y,R_z \} \rangle = \langle R_x,R_y,R_z \rangle \label{so3def} \end{equation} where care must be taken to avoid confusing this language with the similar notation used to denote the span of certain vectors in a vector space. We interpret (\ref{so3def}) as meaning that any matrix in $\SO(3)$ can be written as the product of matrices of the form $R_x$, $R_y$, $R_z$; there is no restriction on how many matrices might be needed, nor on what parameter values (rotation angles) are allowed.