Is the dual of a cone can be defined by the dual of the closure of its convex hull?

Question

Let $K$ be a nonempty cone in $\mathbb{R}^n$. We denote the dual of a cone $K$ as $K^*$. Show that $$K^*=(\mathop{\boldsymbol{cl}} \mathop{\boldsymbol{conv}}K)^*$$

How can we describe the closure of the convex hull of a cone mathematically (using algebra)? The proof seems hard to start.