Here's a proposition from Lee's Smooth Manifolds:
Let $V$ be a real vector space of dimension $n$, let $(E^i)$ be any basis for $V$, and let $(\epsilon^i)$ be the dual basis. The set of all $k$-tensors of the form $\epsilon^{i_1}\otimes...\otimes\epsilon^{i_k}$ for $1 \leq i_1,...,i_k \leq n$ is a basis for the set of all covariant $k$-tensors on $V$, denoted by $T^k(V)$, which therefore has dimension $n^k$.
I want to construct an example based on this proposition. Let $V=\mathbb{R}^3$, and let $e_1, e_2, e_3$ denote the usual basis for $\mathbb{R}^3$. Then the dual basis $(\epsilon^i)$ is $dx, dy$ and $dz$. Then $dx\otimes dy, dx\otimes dz, dy\otimes dz, dy\otimes dx, dz\otimes dy, dz\otimes dx, dx\otimes dx, dy\otimes dy$ and $dz\otimes dz$ form a basis for $T^2(\mathbb{R}^3)$.
Is the example correct until this point? If so, which of them are the symmetric 2-tensors?