Let $k>0$ be the dimension of the range of $f$. We can then write $\mathbb R^{n}=\ker f\oplus V$ with $\dim\ker f=n-k,\dim V=k$. Choose a basis $(e_{i})_{i=1}^{n}$ of $\mathbb R^{n}$ with $(e_{i})_{i=1}^{k}\subset V,\ (e_{j})_{k+1}^{n}\subset\ker f$. Now the basis $(e_{i})_{i=1}^{k}\cup(e_{1}+e_{j})_{j=k+1}^{n}$ satisfies the requirements.