Showing that $\mathbb{F}^n = \mathrm{span}\{\sum_{i=1}^ne_i\}\oplus \mathrm{span}\{\sum_{i=1}^nc_ie_i\mid \sum_{i=1}^nc_i = 0\}$

Question

Let $\mathbb{F}^n$ be a vector space over the field $\mathbb{F}$ with a basis $\{e_1,\dots,e_n\}$. I'm trying to show that $\mathbb{F}^n = \mathrm{span}\{\sum_{i=1}^ne_i\}\oplus \mathrm{span}\{\sum_{i=1}^nc_ie_i\mid \sum_{i=1}^nc_i = 0\}$ as a part a direct proof (as suggested by Qiaochu) for the claim discussed here: Irreducibility of the standard representation of $S_n$. . Currently, I'm stuck at the following: Let $v \in \mathbb{F}^n, v = \sum_{i=1}^nc_ie_i, c_i\in \mathbb{F},i=1,\dots,n$. Then if $v = \sum_{i=1}^nd_ie_i + \beta\sum_{i=1}^ne_i$ with $d_1,\dots,d_n, \beta \in \mathbb{F}, \sum_{i=1}^nd_i = 0$, we have by the linear independence of $e_1,\dots,e_n$ that $c_i = d_i + \beta, \forall i = 1,\dots,n$. And I'm not really sure how to proceed onwards from here. The equation implies that $c_i - d_i = c_j - d_j, \forall i, j = 1,\dots,n$, which feels a bit shady to me.