Prove that any $n \times n$ orthogonal matrix $A$ is a product of $n-1$ Householder reflections and diagonal matrices of $1$ except at one diagonal that is $-1$ (i.e. reflections about $\mbox{span}\{e_j\}^\bot$ for some $j$).
There is a hint that asks "what can be said about an upper triangular matrix that is orthogonal?" I know that an upper triangular matrix that is orthogonal is a diagonal matrix, but am unsure as to how this helps with the proof.