Let $V$ be an $n$-dimensional $\mathbb{K}$-vector space. By a simple calculus trick (*) on homogeneous functions of degree $n$ the determinant is a linear map on the $n$-th symmetric power of the vector space of endomorphisms: $$det\,:\;S^n(End(V))\rightarrow\mathbb{K}\;.$$ Q: Is there an explicit formula? Reference?

It seems I have one in terms of the skew Hopf algebra convolution product of homomorphisms of the exterior algebra, which looks symmetric on $0$-degree homogeneous homomorphisms... (Being a left-right confuser I would have to check all the signs 3 times over. Plus, it looks suspiciously pretty and suggests benefits of skew Hopf algebra in the theory of determinants, so it should be in a good multilinear algebra textbook if my hunch is right. I'm not interested in quantum groups or braided monoidal category theory. Plain Hopf algebra is already interesting enough, as I once found in tensor calculus.)

(*) Serge Lang, Differential and Riemannian Manifolds (1995), p.7

Let $V$ be an $n$-dimensional $\mathbb{K}$-vector space. By a simple calculus trick (*) on homogeneous functions of degree $n$ the determinant is a linear map on the $n$-th symmetric power of the vector space of endomorphisms: $$\det\,:\;S^n(End(V))\rightarrow\mathbb{K}\;.$$ Q: Is there an explicit formula? Reference?

It seems I have one in terms of the skew Hopf algebra convolution product of homomorphisms of the exterior algebra, which looks symmetric on $0$-degree homogeneous homomorphisms... (Being a left-right confuser I would have to check all the signs 3 times over. Plus, it looks suspiciously pretty and suggests benefits of skew Hopf algebra in the theory of determinants, so it should be in a good multilinear algebra textbook if my hunch is right. I'm not interested in quantum groups or braided monoidal category theory. Plain Hopf algebra is already interesting enough, as I once found in tensor calculus.)

(*) Serge Lang, Differential and Riemannian Manifolds (1995), p.7