The symmetric algebra of an $R$-module $M$ is defined to be the quotient of the tensor algebra $\mathcal{T}(M)$ by the ideal $C(M)$ generated by all elements of the form $m_1\otimes m_2-m_2\otimes m_1$ for all $m_1,m_2\in M$, and it is denoted by $S(M)$.
A theorem in Dummit and Foote claims that the $k$th symmetric power $S^k(M)$ of $M$ is equal to $M\otimes\cdots\otimes M$ (k factors) modulo the submodule generated by all elements of the form $(m_1\otimes\cdots\otimes m_k)-(m_{\sigma(1)}\otimes\cdots\otimes m_{\sigma(k)})$ for all $m_i\in M$ and all $\sigma\in S_k$.
Now the authors claim that it is "easy to see" that since $\sigma$ can be written as a product of transpositions, every element of the form above can be written as a sum of elements of the form $m_1\otimes\cdots\otimes m_{i-1}\otimes(m_i\otimes m_{i+1}-m_{i+1}\otimes m_i)\otimes m_{i+2}\otimes\cdots\otimes m_k$.
Why is this so?