Is it a simplicial cone always a pointed cone?

Question

I'm trying to solve a problem involving cones and their properties. Considering a simplicial cone (with $n$ edges by definition), $K\subset\mathbb{R}^{n}$, the claim is that $K$ is always a pointed cone in the sense that $K=\{v+\lambda_{1}w_{1}+\lambda_{2}w_{2}+...+\lambda_{m}w_{m}\mid w_{j},v\in\mathbb{R}^{n}; \lambda_{j}\geq 0;1\leq j\leq m \} $. Working out a little bit it is easy to show that $K$ must be generated by $n$ linearly independent vectors, so I concluded that $K$ must be all the space, then I put $K$ generated by all unity vectors in all $2n$ directions of axes, so their nonnegative linear combinations will generate all space. I don't feel confident with this reasoning, anybody could help me to clarify these things? Thanks for any help!