Unique-ish Solution

Let $A$ be a $3 \times n$ matrix for some $n \geq 2$ and $\alpha = \{e_1, e_2, e_3\}$ be the canonical basis of $\mathbb{R}^3$. Suppose that $Ax = e_1$ has no solution and $Ax = e_2$ has a unique solution. Find $n$.