Let $A\in\mathfrak{gl}(\mathbb{R},n)$ be an endomorphism, and think up to conformal factors (in particular, $\Lambda^n\mathbb{R}^n$ will be the same as $\mathbb{R}$). By the total polarization $\widehat{p}$ of a homogeneous polynomial $p\in S^k V$, where $V$ is a vector space, I mean the image of $p$ via the embedding $S^k V\subset V^{\otimes k}$. E.g., if $Q$ is a quadratic form, $B:=\widehat{Q}$ is the associated bilinear form.

"THEOREM" A. There exists a unique unitary graded algebra endomorphism $\varphi_A$ of $\Lambda^\bullet\mathbb{R}^n$ such that $\left.\varphi_A\right|_{\mathbb{R}^n}=A$.

"THEOREM" B. There exists a unique graded derivation $\delta_A$ of degree 0 of $\Lambda^\bullet\mathbb{R}^n$ such that $\left.\delta_A\right|_{\mathbb{R}^n}=A$.

As a corollary, we get $\det\, A= \left.\varphi_A\right|_{\Lambda^n\mathbb{R}^n}$ and $\mathrm{tr}\, A=\left.\delta_A\right|_{\Lambda^n\mathbb{R}^n}$.

QUESTION: what is known about $\widehat{\mathrm{det}}$? Should be a very fundamental object, but I never heard about it. Moreover, it is possible to characterize it intrinsically, in analogy with "THEOREM" A above?

Let me add a few comments. In the case $n=2$ the answer is known (I've just learned it from a very beautiful MO answer---look here: http://mathoverflow.net/a/32583/22606): $\widehat{\mathrm{det}}$ is the Killing form. But what happens as $n$ grows larger? In fact, my question is just the tip of the iceberg, since I may extend it to all the coefficients $\psi_k(A)$ of the characteristic polynomial $p_A(t)$ of $A$, understood as homogeneous polynomials on $\mathfrak{gl}(\mathbb{R},n)$: how to describe their total polarizations?

Of course, I'm able to produce a brutal answer, namely $$ \widehat{\psi_k}:(A_1,\ldots,A_k)\in \mathfrak{gl}(\mathbb{R},n)^{\otimes k}\longmapsto \mathrm{tr}\,(A_1\wedge\cdots\wedge A_k)\in\mathbb{R}^n, $$ but then I'm not able to justify the symmetry of $\widehat{\psi_k}$, and the very definition of $A_1\wedge\cdots\wedge A_k$, which should be an endomorphism of $\Lambda^k\mathbb{R}^n$, is unclear. My guess is that the last formula can be made "clean" by combining the above defined $\varphi_{A_i}$'s and $\delta_{A_i}$'s, but guessing is not enough, and it's all I can do!