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.

I agree with SmileyCraft. In finite dimensional vector spaces, bilinear transformations, as linear transformations, can be written in terms of the values that they adopt in a given base:$$\left \langle x,y \right \rangle=\sum_{i,j=1}^{n}x_iy_j\left \langle e_i,e_j \right \rangle.$$
I believe you can arrive in this representation without difficult, proving then you suspicion.
