There exist the famous theorem about a basis for dual space

Let $\mathbb V$ be finite dimensional vector space over $F$ and $\mathcal{B} = \{\alpha_1, \ldots ,\alpha_n\}$ is basis for vector space $\mathbb V$ then $\mathcal{B^*} = \{f_1, \ldots ,f_n\}$ is basis for dual space $\mathbb V^*$ such that $f_i(\alpha_{j})=\delta_{ij}$

Is the converse of this Theorem true, to explain more:

Let $\mathcal{B^*} = \{f_1, \ldots ,f_n\}$ be basis for dual space $\mathbb V^*$. Does there exist $\mathcal{B} = \{\alpha_1, \ldots ,\alpha_n\}$ such that $\mathcal{B}$ would be basis for vector space and $f_i(\alpha_{j})=\delta_{ij}$?