You are basically looking for trivial $O(\mathbb{C}^n)$ representations contained in $\Lambda^k \mathfrak{p}^*.$ Since your representation $\mathfrak{p}$ is the space of symmetric matrices, you can find the whole decomposition into irreducibles of $\Lambda^k S^2 \mathbb{C}^n$ as a special case of plethysm for the Lie group $O(\mathbb{C}^n).$ In fact, this very special case of plethysm was already dealt with on this site for the group $GL(\mathbb{C}^n)$ in (at least) two questions:

Known decomposition of $\bigwedge^k Sym^d \mathbb C^n$ in special cases?

What is known about this plethysm?

Please note, that irreducible $GL(\mathbb{C}^n)$ representations decompose further under restriction to $O(\mathbb{C}^n)$ -- see e.g. Symmetry, Representations, and Invariants by Nolan Wallach and Roe Goodman.

To check whether you get correct decompositions and/or multiplicities you can use mathematical software. See e.g.:

Symmetric powers of Schur polynomials

You are basically looking for trivial $O(\mathbb{C}^n)$ representations contained in $\Lambda^k \mathfrak{p}^*.$ Since your representation $\mathfrak{p}$ is the space of symmetric matrices, you can find the whole decomposition into irreducibles of $\Lambda^k S^2 \mathbb{C}^n$ as a special case of plethysm for the Lie group $O(\mathbb{C}^n).$ In fact, this very special case of plethysm was already dealt with on this site for the group $GL(\mathbb{C}^n)$ in (at least) two questions:

Known decomposition of $\bigwedge^k Sym^d \mathbb C^n$ in special cases?

What is known about this plethysm?

Please note, that irreducible $GL(\mathbb{C}^n)$ representations decompose further under restriction to $O(\mathbb{C}^n)$ -- see e.g. Symmetry, Representations, and Invariants by Nolan Wallach and Roe Goodman.

To check whether you get correct decompositions and/or multiplicities you can use mathematical software. See e.g.:

Symmetric powers of Schur polynomials

I've whipped up some Sage code which however seems to be quite slow: https://cocalc.com/share/25e624ba-9091-484f-af2e-71deb7120f58/trivial%20reps%20in%20exterior%20algebra.sagews?viewer=share

You are basically looking for trivial $O(\mathbb{C}^n)$ representations contained in $\Lambda^k \mathfrak{p}^*.$ Since your representation $\mathfrak{p}$ is the space of symmetric matrices, you can find the whole decomposition into irreducibles of $\Lambda^k S^2 \mathbb{C}^n$ as a special case of plethysm for the Lie group $O(\mathbb{C}^n).$ In fact, this very special case of plethysm was already dealt with on this site for the group $GL(\mathbb{C}^n)$ in (at least) two questions:

Known decomposition of $\bigwedge^k Sym^d \mathbb C^n$ in special cases?

What is known about this plethysm?

Please note, that these decomposition into $GL(\mathbb{C}^n)$ representations decompose further under restriction to $O(\mathbb{C}^n)$ -- see e.g. Symmetry, Representations, and Invariants by Nolan Wallach and Roe Goodman.

To check whether you get correct decompositions and/or multiplicities you can use mathematical software. See e.g.:

Symmetric powers of Schur polynomials

You are basically looking for trivial $O(\mathbb{C}^n)$ representations contained in $\Lambda^k \mathfrak{p}^*.$ Since your representation $\mathfrak{p}$ is the space of symmetric matrices, you can find the whole decomposition into irreducibles of $\Lambda^k S^2 \mathbb{C}^n$ as a special case of plethysm for the Lie group $O(\mathbb{C}^n).$ In fact, this very special case of plethysm was already dealt with on this site for the group $GL(\mathbb{C}^n)$ in (at least) two questions:

Known decomposition of $\bigwedge^k Sym^d \mathbb C^n$ in special cases?

What is known about this plethysm?

Please note, that irreducible $GL(\mathbb{C}^n)$ representations decompose further under restriction to $O(\mathbb{C}^n)$ -- see e.g. Symmetry, Representations, and Invariants by Nolan Wallach and Roe Goodman.

To check whether you get correct decompositions and/or multiplicities you can use mathematical software. See e.g.:

Symmetric powers of Schur polynomials