Explicit matrix representation of an algebraic extension

Question

This may be considered an extension of this question. Let $\mathbb{F}$ be a field, and let $p(X)\in\mathbb{F}[X]$ be an irreducible polynomial. Let $\mathbb{F}_p$ be the extension of $\mathbb{F}$ by adding the roots of $p$. Can we find an explicit matrix representation of $\mathbb{F}_p$ in $\mbox{End}_\mathbb{F}\mathbb{F}_p\simeq\mbox{gl}(\mathbb{F},[\mathbb{F}_p:\mathbb{F}])$ in terms of coefficients or roots of $p(X)$? I think $[\mathbb{F}_p:\mathbb{F}]=(\deg p-1)!$ can be shown by induction, but I find it hard to establish a tractable representation similar to the well known one for $\mathbb{C}$ over $\mathbb{R}$. I am much afraid of clever things like Galois or so, and I would appreciate an answer in simpler terms. Thank you.