Let $$\det\nolimits_n=\sum_{\pi\in\mathfrak{S}_n} \operatorname{sgn}(\pi)\cdot x_{1,\pi(1)}\cdots x_{n,\pi(n)} \in \mathbb{C}[x_{ij}\mid 1\le i,j\le n]$$ be the determinant polynomial. It defines a singular hypersurface $Z_n\subseteq \mathbb{C}^{n\times n}$ consisting of the rank-deficient complex $n\times n$ matrices.
Since we are in characteristic zero, there is an embedded resolution $\phi_n: Y_n\to \mathbb{C}^{n\times n}$ such that the strict transform of $Z_n$ is a nonsingular subvariety of $Y_n$.
My question is, has anyone ever given an explicit description of $\phi_n$, in the sense that $\phi_n$ is expressed as a (sequence of) blow-up(s) along something that possibly has some nice description? Any reference is more than welcome.
This isn't an embedded resolution but it is at least quite a nice resolution so maybe it will be helpful.
This will work more generally for any determinantal variety so I will just talk about it in that generality. Let $M_{m,n} = \mathbb{C}^{mn}$ viewed as $m \times n$ matrices with $n \geq m$.
Then take $Z_{m,n} \subset M_{m,n}$ be the vanishing of all maximal minors of the generic $m \times n$ matrix. We have the incidence variety
$$ X_{m,n} = \{([v],M) : vM = 0\} \subset \mathbb{P}^{m - 1} \times M_{m,n} $$
where $[v]$ is the class of an $m$-vector $v$ in projective space. Then the restriction of the second projection $pr_2 : \mathbb{P}^{m-1} \times M_{m,n} \to M_{m,n}$ to $\alpha: X_{m,n} \to Z_{m,n}$ is a resolution of singularities.
This is discussed for example here.