Generalized "scalar product" based on multilinear form? In an $\mathbb{R}$-vector space $V$, the scalar product is a paradigmatic example of a non-degenerate, symmetric, positive-definite bilinear form $\beta : V \times V \to \mathbb{R}$.
I wonder if the scalar product can be generalized to more than two vectors. I am particularly eager to know if an operation analogous to the scalar product can be defined, which is, however, based on an $m$-multilinear form $\mu : V^m \to \mathbb{R}$ with $m > 2$. If so, what could be the geometrical interpretaion of such a "multilinear scalar product"?
Edit: If possible, the properties of the ordinary scalar product should be preserved.
 A: Let me explain supposing that $m=2$ first.
If you consider the set of all bilinear maps $V\times V\to\Bbb R$, 
with $V={\rm span}\{b_i\}$, one should also consider that the effect of taking a pair of basic duals $\beta^j$ such that $\beta^j({b_i})=\delta^j{}_i$. 
With pair of $\beta^i,\beta^j$ one gets
$$\beta^i\otimes\beta^j:V\times V\to\Bbb R$$
given by
$$\beta^i\otimes\beta^j(v,w)=\beta^i(v)\beta^j(w).$$
This map is bilinear. It is used to forge out all bilinear maps into a vector space generated by $\{\beta^i\otimes\beta^j\}$, 
could we dubbed this space as $V^*\otimes V^*$?
So, this gives us an representation of a bilinear map 
$T:V\times V\to\Bbb R$ in the style
$$T=T_{sr}\beta^s\otimes\beta^r$$
One can then take advantage of a certain interior product on  $V$ are given.
That is, if an inner product $\bullet:V\times V\to\Bbb R$ is given one forms its metric tensor (matrix) 
$$G=[g_{ij}],$$
where $g_{ij}=b_i\bullet b_j$. This construction allow to calculate for any $v,w\in V$:
$$v\bullet w=v^sv^rg_{sr}.$$
With this idea one define for $f,\phi$ a pair of covectors 
$$f\bullet\phi=f_s\phi_rg^{sr}$$
where $g^{sr}$ are entries of the matrix $G^{-1}$.
Now for a pair of contravariant tensors $T=T_{sr}\beta^s\otimes\beta^r$
and $U=U_{sr}\beta^s\otimes\beta^r$, one can guess that a construction like
$$T\bullet U:=T_{ij}U_{kl}g^{ik}g^{jl}$$ 
is useful as an analog of a inner product but now in $V^*\otimes V^*$.
Finally  for the $m$-multilinear maps $V^m\to\Bbb R$, one is compelled
to, in the moment of consider two of them, say, 
$$\mu=\mu_{i_1...i_m}\beta^{i_1}\otimes\cdots\otimes\beta^{i_m},$$ 
and
$$\theta=\theta_{i_1...i_m}\beta^{i_1}\otimes\cdots\otimes\beta^{i_m},$$
to take as their inner product the number
$$\mu\bullet\theta=\mu_{i_1...i_m}\theta_{j_1...j_m}g^{i_1j_i}\cdots g^{i_mj_m}.$$
