Properties of Directional Derivative

Any vector $\vec v$ with initial point $a\in\mathbb R^n$ yields a map D_\vec v|_a:C^\infty(\mathbb R^n)\rightarrow \mathbb R, which takes the directional derivative in the direction $\vec v$ at $a$ : D_\vec v|_af=D_\vec vf(a)=\lim_{t\to 0}\frac {f(a+t\vec v)-f(a)}t. This operation is linear over $\mathbb R$ and satisfies the product rule: D_\vec v|_a(fg)=f(a)D_\vec v|_ag+g(a)D_\vec v|_af. Conversely, if a map $\omega:C^\infty(\mathbb R^n)\rightarrow \mathbb R$ is linear over $\mathbb R$ and satisfies the following product rule: $\omega(fg)=f(a)\omega g+g(a)\omega f,$ must $\omega$ be a directional derivative in certain direction at $a$ ?