Trigonometrical algebras

Euclidean $n$-dimensional spaces that have an analog of a vector product, i.e., a bilinear binary operation satisfying the identity $|x\cdot y|^2+(x,y)^2=|x|^2\cdot|y|^2$ ($(\cdot,\cdot)$ is a scalar product). It is clarified for which $n$ such a product exists.