Does there exist a unique definition of dot product in $\mathbb R^n$ such that the standard basis is orthonormal? Does there exist a unique definition of scalar product in $\mathbb R^n$ so that the standard basis is orthonormal? I can't find a definition of dot product different from the usual definition. Is this true or false?. Thanks!
Edit
Let $\langle ,\rangle_1$ y $\langle ,\rangle_2$ be two different dot products such that the standard basis is orthonormal with those dot products. Both dot products are associated to Gram matrices $G, G'$ respectively, but since they make the standard basis orthonormal,
  necessarily $G = G' = I_n$. Hence dot products are the same since given two vectores $x,y\in\mathbb R^n$ we have that $x^t G
  y = x^t G' y = x^t y $. Sorry for my english, it is terrible. Is this valid? Thanks again!
 A: Once you define the dot product on a basis, you define it everywhere by linearity.  So the standard dot product you are used to is in fact the only dot product where the standard basis is orthonormal
A: Every symmetric bilinear form can be expressed as a symmetric matrix $B$ so that $B(x,y)=xBy^T$ with usual matrix multiplication. I'm abusing notation by using $B$ for both $B(-,-)$ and the matrix $B$, but I hope it is not too distracting.
If you want want the standard basis to be "orthonormal" in the sense that the elements are pairwise orthogonal with $B(e_i,e_i)=1$, then for that to happen, you would need two things to happen. 
Firstly $B(e_i,e_i)=e_iBe_i^T=1$. If you look at what this means, it says that the diagonal elements of the matrix $B$ are all 1.
Secondly, $B(e_i,e_j)=e_iBe_j^T=0$ for $i\neq j$. If you look at what this means, it says that $B_{ij}=0$ for all $i\neq j$. So that means that $B$ has to be the identity matrix, and the bilinear form is in fact the usual dot product: $xI_ny^T=xy^T$.

You might be interested in knowing that every symmetric bilinear form is equivalent to one with a diagonal matrix whose diagonal entries can be whatever mixture of elements from $\{0,-1,1\}$ that you like. This includes bilinear forms $A$ for which $A(b,b)=-1$ for some basis elements, and $A(b,b)=0$ for some basis elements.
These are quite a bit different from the usual dot product because they are not positive definite, and have vectors with "zero length," but I can assure you that they have their uses. Some examples are the bilinear forms with $(1,-1,-1,-1)$ or $(-1,1,1,1)$ on the diagonal, which can be used on $\Bbb R^4$ to model spacetime.
