I understand a rotation should preserve length and angle and hence the dot product. Since anything that preserves the dot product is a linear transformation, then a rotation can be represented by a matrix. Let's only consider real matrices.
However, there is something else that a rotation preserves and I am here to ask what it is. A linear transformation that preserves length must be orthogonal, however, an orthogonal matrix is a rotation if its determinant is $1$. That means a rotation is a special case of orthogonal matrix.
So what makes a rotation special from orthogonal matrices? What else does a rotation preserve in addition to length and angle? Anyone can provide a prove that a matrix is a rotation iff it is an orthogonal matrix with determinant $1$?