Map sending $k$-linear functions in $V_1\times V_2\times\cdots \times V_n$ into linear maps in $V_1 \otimes V_2 \otimes \cdots\otimes V_n$

Question

Say we have an n-linear map $L_n$ (meaning linear in each of the n -variables) from

$ V_1 \times V_2 \times.....\times V_n$ Vector spaces over the same base field K , into, say,

the Reals. A result is that in the vector space :

$$ V_1 \otimes V_2 \otimes....\otimes V_n $$ , there is a map $L_n'$ which

is linear and which factors through $L_n$ (Sorry, don't know how to draw

diagrams). Just what is this map sending n-linear functions $L_n$ into linear

ones in the tensor product ? And what is the representation of the linear map

on the tensor product,

as, say, a matrix? For simplicity, please use if possible, $V_i$ as the

$\mathbb R^n$ for some $n$ with the standard basis $e_i= \delta_i^j$.

I would also like to know if the expression $a \otimes b$ describes the

equivalence class of a bilinear map that sends the pair $(a,b)$ to $1$ ( and,

equivalently, $c(a \otimes b )$ represents the class of bilinear maps taking

the pair $(a,b)$ to $c$.

I assume this has to see with maps factoring through the kernel, but I am

having trouble pinning things down more carefully.

Thanks.