Given a finite dimensional vector space $V$, how do elements of $\text{Sym}^2(V)$ give a map $V^*\rightarrow V$? An article I'm reading claims that if $V$ is a finite dimensional vector space over a field $k$, and $f\in \text{Sym}^2(V)$, then $f$ induces a morphism $V^*\rightarrow V$, where $V^*$ denotes the dual space.
If we let $x_1,\ldots,x_r$ be a basis of $V$, then we can write $f = \sum_{i\le j}a_{ij}x_ix_j$, and given $\varphi\in V^*$, one could try to consider the map
$$\varphi\mapsto \sum_{i\le j}a_{ij}\varphi(x_i)x_j\in V$$
but surely this isn't well defined...
EDIT: This is the bottom of page 38 of the stacks project.

Define $f:V\otimes_KV\to\mathrm{Hom}_K(V^*,V)$ by
$$
  f(v\otimes w)(\phi)=\phi(v)w+\phi(w)v.
$$
This is clearly symmetric under $v\otimes w\mapsto w\otimes v$, so it factors to a map $\mathrm{Sym}^2(V)\to\mathrm{Hom}_K(V^*,V)$.
As Lord Shark the Unknown comments, the source link gives a matrix for the transformation using a basis for $V$. This can be verified to agree with the above.
