Let $V$ be an $n$-dimensional (real) vector space equipped with a (real-valued) symmetric bilinear form, denoted $\langle \Bbb{u}, \Bbb{v} \rangle$ for $\Bbb{u}, \Bbb{v} \in V$. This bilinear form induces a "standard" rank 2 symmetric tensor $\varphi_2 \in \mathrm{Sym}^2 \big( \bigwedge^2V \big)$ defined by
\begin{equation} \varphi_2 \big((\Bbb{a} \wedge \Bbb{b}) \cdot (\Bbb{c} \wedge \Bbb{d}) \big) \ := \ \det \begin{pmatrix} \langle \Bbb{a} , \Bbb{c} \rangle & \langle \Bbb{a} , \Bbb{d} \rangle \\ \langle \Bbb{b} , \Bbb{c} \rangle & \langle \Bbb{b} , \Bbb{d} \rangle \end{pmatrix} \end{equation}
where $\Bbb{a} \wedge \Bbb{b}$ and $\Bbb{c} \wedge \Bbb{d}$ are two decomposable anti-symmetric 2-forms taken from $\bigwedge^2V$.
Question: For $k \geq 3$ is there a standard choice of rank $k$ symmetric form $\varphi_k \in \mathrm{Sym}^k \big( \bigwedge^2 V \big)$ induced from the intial bilinear form $\langle \Bbb{u} , \Bbb{v} \rangle$ on $V$ ?
thanks, ines.