Suppose $M,N$ are open subsets of $\R^n$. A smooth map $F:M\rar N$ is conformal iff
$$
\forall x\in M\,\forall u,v\in\R^n:\quad
\la DF(x)u,DF(x)v\ra=h(x)^2\la u,v\ra
$$
Fix $x$; the mapping $B:(u,v)\mapsto\la DF(x)u,DF(x)v\ra$ is bi-linear and symmetric. By the polarization formula we conclude that $B(u,v)=h(x)^2\la u,v\ra$ if and only if $B(u,u)=h(x)^2\la u,u\ra$.