Let $E$ be an euclidean space (over $\mathbb{R}$), I have to prove that every hyperplane of the linear maps over $E$ contains an orthogonal map (or equivalently, matrix).
What I've tried doing is saying that any such hyperplane can be written as the orthogonal space of a particular map. Then, I'm stuck: I've tried the case where this matrix is symmetric (and therefore can be written as a diagonal matrix over an orthonormal basis) in order to imitate the proof where you have to find an invertible matrix and not an orthogonal one.
I'm not sure where to go from here...
The hyperplane can be written as the orthogonal of a particular matrix for the scalar product $<A,B> \mapsto Tr(^tA B)$.
If the matrix is symmetric, then it can be written as $A=^tP\Delta P$ where $P$ is orthogonal and $\Delta$ is diagonal. Let $M$ be a matrix containing four blocks, from left to right and top to bottom: zeroes, $I_{n-1}$, 1, zeroes.
It is indeed orthogonal and the $<\Delta,M>=0$ therefore $^tPMP$ is an orthogonal matrix in the given hyperplane.
In the general case, any matrix can be written as the product of an orthogonal matrix and a symmetric matrix $OS$. If $M$ is in the hyperplane orthogonal to $S$, then $^tOM$ is indeed symmetric and contained in the hyperplane orthogonal to $OS$.