Let $\text{char}\,k = 0$ and $n \ge 2$. What is the easiest way to see that $k[x_1, \dots, x_n]$ is a free $k[x_1, \dots, x_n]^{S_n}$-module with basis$$x_2^{m_2}x_3^{m_3} \dots x_{n-1}^{m_{n-1}} x_n^{m_n},\text{ }m_2 \in [0, 1],\text{ }m_3 \in [0, 2], \dots,\text{ }m_n \in [0, n-1]?$$