Why does an orthogonal matrix have to be square? I understand intuitively why this has to be the case (otherwise you could lose a dimension / gain a dimension which changes the length), but what is the formal proof that an orthogonal matrix has to be square? 

Just to sum up the comments, your book says a linear transformation $T:\mathbb R^n\to\mathbb R^n$ is orthogonal if it preserves the length of vectors. The matrix of a transformation from a vector space to a vector space of the same dimension is necessarily square, so this is baked into the definition of an orthogonal matrix. 
If the book said "A linear transformation $T:\mathbb R^n\to\mathbb R^n$ is said to be blah if it blahs", you'd still know that its matrix is square. 
