Let $n\geq 2$ be an integer and $p$ a prime with $ p>n $. Let $ \mathfrak{sl}_{n}(\mathbb{F}_{p})$ denote the special linear Lie algebra over a finite field $ \mathbb{F}_{p} $, i.e. the space of $ n\times n $ matrices with trace zero over $ \mathbb{F}_{p} $ with the Lie bracket $ [X,Y]=XY-YX $. Note that our assumption implies that $ \mathfrak{sl}_{n}(\mathbb{F}_{p}) $ is a simple Lie algebra over $\mathbb{F}_p$.
Question: Given a family $\{b_{ij}\}_{i,j\in I} $ of elements in $ \mathbb{F}_{p} $ where $ I=\{1,\cdots,n^{2}-1\} $ and $ b_{ij}=0 $ if $ i=j $. Suppose that at least one of $ b_{ij},b_{ji} $ is non-zero for all $ i\neq j $. Is it true that there is no basis $ \{A_{i}\}_{i\in I} $ of $ \mathfrak{sl}_{n}(\mathbb{F}_{p}) $ such that $$ \sum_{j\neq i}b_{ij}[A_{i},A_{j}]=0 ?$$ When $n=2$, we find that $ [A_{1},A_{2}],[A_{1},A_{3}],[A_{2},A_{3}] $ must be linearly independent since $ \mathfrak{sl}_{2}(\mathbb{F}_{p}) $ is a simple Lie algebra. From this, it's easy to see that the answer to the question is Yes. But I have no idea for $n\geq 3$. Any comments and reference would be highly appreciated.