So say I have a real vector space $V$ of dimension $n$. Consider the tensor space $T^{(2, 0)}(V) = V \otimes V$, note that $V \otimes V$ is isomorphic to the vector space of multilinear maps from $V^* \times V^* \to \mathbb{R}$, that is $$T^{(2, 0)}(V) = V \otimes V \cong L\left(V^*, V^*; \mathbb{R}\right)$$
So any element $\alpha \in V \otimes V$ is of the form $\alpha = \mathfrak{a} \otimes \mathfrak{b}$ where $\mathfrak{a}$ and $\mathfrak{b}$ are elements of the double dual $V^{**}$ (under an isomorphism to $V$) and $\mathfrak{a} \otimes \mathfrak{b} : V^* \times V^* \to \mathbb{R}$ is a multilinear map defined by $(\mathfrak{a} \otimes \mathfrak{b})(f, g) = \mathfrak{a}(f)\cdot \mathfrak{b}(g)$
Now consider the tensor space $T^{(0, 2)}(V) = V^* \otimes V^*$. Similarly we have that $$T^{(0, 2)}(V) = V^* \otimes V^* \cong L\left(V, V; \mathbb{R}\right)$$
So any element $\beta \in V^* \otimes V^*$ is of the form $\beta = \omega \otimes \eta$ where $\omega,\eta \in V^*$ and $\omega \otimes \eta : V \times V \to \mathbb{R}$ is a multi-linear map such that $(\omega \otimes \eta)(x, y) = \omega(x)\cdot \eta(y)$.
Firslty is everything above that I've said correct? If so what if we have the mixed tensor space $T^{(2, 1)}(V) = V \otimes V \otimes V^*$, is this tensor space isomorphic to $L(V^*, V^*, V; \mathbb{R})$? In other words is every element $\kappa \in V \otimes V \otimes V^*$ of the form $\kappa = \mathfrak{a}\otimes \mathfrak{b} \otimes \omega$ where $\mathfrak{a}, \mathfrak{b}$ are elements of $V$ under an isomorphism to $V^{**}$ and $\omega \in V^*$ and $$\kappa = \mathfrak{a}\otimes \mathfrak{b} \otimes \omega : V^* \times V^* \times V \to \mathbb{R}$$ is defined by $$(\mathfrak{a}\otimes \mathfrak{b} \otimes \omega)(f, g, x) = \mathfrak{a}(f) \cdot \mathfrak{b}(g)\cdot \omega(x)$$
Is the above correct?
EDIT: Apart from my error in writing everything as simple tensors, are my above ideas correct?