Fundamental weights of $A_n$

Question

I have the following problem: Let $\mathfrak{g}$ be the Lie algebra of type $A_n$. We choose $e_i^*-e_{i+1}^*$ as simple roots. Is there a closed formula for the fundamental weights? Thank you!

Answer 1

If I understand the notation well, the $i$th fundamental weight should be $e_1^*+\dots+e_i^*$.