Confusion over this definition of a tensor

A tensor is defined as an object that linearly maps an ordered pair of vectors from the Cartesian product $V\times V$ to scalars, where both vectors in the ordered pair belong to $V$. Hence \begin{equation}\tag{1} \bf{T}: \it{V\times V \rightarrow} \mathbb{R} \end{equation} Let $v$ and $w$ be vectors that belong to $V$ and $(v,w)$ belong to $V\times V$. Then $$\bf{T} \it{(v,w)}=a$$ Now if we represent $\bf{T}$ as the tensor product of dual space vectors, $v^*\otimes w^*$ then the above statement is written as (this equation is the definition of tenosr product) $$\langle v^*,v\rangle \langle w^*,w\rangle =a$$ Hence $a$ is simply $v^*_\mu v^\mu w^*_\nu w^\nu$. The representation of tensor as the tensor product of dual space vectors helps us express this map in terms known objects. Hence a tensor is expressed as $$\bf{T}=\it{v^*_\mu w^*_\nu e^\mu \otimes e^\nu}$$ Equation $(1)$ is what the author meant when he said that tensor eats a bunch of vectors and spits out a scalar.