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I'll propose to you another (slightly different, but isomorphic) definition of the adjugate (classical adjoint). Im borrowing from section 8 of http://people.reed.edu/~jerry/332/27exterior.pdf . Let $f:V\rightarrow V$ (with $n$ the dimension of $V$). We have a canonical isomorphism $\phi:V=\wedge^1 V\rightarrow\mathrm{Hom}(\wedge^{n-1} V,\wedge^n V)$ ...
Write the Kronecker Delta as $$\delta_{ij}=\hat x_i \cdot \hat x_j \tag 1$$ in terms of then inner product of Cartesian unit vectors. Write the Levi-Civita symbol as $$\epsilon_{ijk}=\hat x_i\cdot(\hat x_j \times \hat x_k) \tag 2$$ in terms of the scalar triple product of Cartesian unit vectors. Using $(2)$, we have \begin{align} ... 1 As I see it, you have \mathbb{R}^{n} along with a group action of S_{n} acting by permuting coefficients of your vectors (wrt lets say the standard basis). For each permutation \pi, you have a map \varphi_{\pi}:\mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R} by \varphi_{\pi}:(x,y) \mapsto x \cdot y^{\pi} (each of these is actually a ... 1 With block matrix notation, where 0_n is the n \times n zero matrix and I_n is the n \times n identity matrix, it's simply the matrix product \omega_p(u,v)= (u_1,\dots,u_{2n}) \begin{pmatrix} 0_n & I_n \\ -I_n & 0_n \end{pmatrix} \begin{pmatrix} v_1 \\ \vdots \\ v_{2n} \end{pmatrix} . $$To see this, call the basis vectors$$ ...