Let $x$ and $y$ be two vectors and $A$ the angle between them. Then we have the scalar product $$x\cdot y = \|x\|\|y\| \cos A$$
Let $x$, $y$ and $z$ be three vectors; $A$ angle between $x$ and $y$; $B$ angle between $x$ and $z$; and $C$ angle between $y$ and $z$. What is the value of the scalar product for the three vectors? Generalization: What is the value of the scalar product for $N$ vectors in $n$-dimensional space?
In 2-dimensional space we define a symmetric bilinear form for scalar product. In n-dimensional space can we define a symmetric multilinear form for N vectors?
The notion of scalar product is defined for 2 vectors, not only for 2-dimensional spaces but for n-dimensional spaces as well, as is the corresponding (for the geometric interpretation) notion of the angle, which also only takes two vectors.
A generelisation in linear algebra is a scalar product that is defined as a bilinear, symmetric (as you said) and positive definite (i.e. $\left<v,v\right>>0 $ for $v\neq0$) map $V\times V \rightarrow \mathbb{K}$ where $\mathbb{K}$ is either $\mathbb{C}$ or $\mathbb{R}$ and $V$ is a vector space.
Now, independent of the dimension of your space you can define multilinear maps which are a generalisation of bilinear maps and can also have properties like symmetry, but scalar product is a term that refers to maps that takes 2 arguments in all occasions I've encountered so far.