Other inner products for $\mathbb{R}^n$ For $\mathbb{R}^n$, the standard inner product is the dot product. It is defined as $ \langle v,\,w\rangle = \sum_i v_i \cdot w_i $. I am aware that any scaled version, namely
$  \langle v,\,w\rangle  = \sum_i\lambda_i\cdot v_i \cdot w_i $ will still satisfy the 4 inner product requirements.
Is there any inner product for  $\mathbb{R}^n$ that is not just a scaled version of the standard dot product?
I tried for  $\mathbb{R}^2$ with $ \langle v,\,w\rangle  =  v_1 \cdot w_2 + v_2 \cdot w_1 $ but that is not positive definite.

It is well known (and easy to prove) that any two finite dimensional inner product spaces are isometrically isomorphic. Hence $ \langle x, y \rangle'$ is an inner product on $\mathbb  R^{n}$ iff there is a vector space isomorphism $T: \mathbb  R^{n} \to \mathbb  R^{n}$ such that $\langle x, y \rangle' =\langle Tx, Ty \rangle$ for all $x,y$. 
