I like the following description of the spectrum of $S^{2m-1}$ via representation theory of $SO(2m)$.
Assume $n=2m-1$.
Let $\mathcal E_0=\{0\}$ and $$\mathcal E_p=\{\lambda_{k,p}:=k^2+k(2m-2)+(p-1)(2m-1-p): k\in \mathbb N\},$$ for $1\leq p\leq m$.
One can check that $\mathcal E_p\cap\mathcal E_{p+1}$ is empty for every $0\leq p\leq m-1$.

The eigenvalues of the Hodge-Laplace operator $\Delta_p$ on $p$ forms on $S^{2m-1}$ are $\mathcal E_p\cup\mathcal E_{p+1}$ for $0\leq p\leq m-1$.
The multiplicity is given by
$$
\textrm{mult}(\lambda) =
\begin{cases}
\dim \pi_{k,p} & \text{ if } \lambda=\lambda_{k,p},\\
\dim \pi_{k,p+1} & \text{ if } \lambda=\lambda_{k,p+1}.
\end{cases}
$$

Here, $\pi_{k,p}$ denotes the irreducible representation of $SO(2m)$ having highest weight $(k\varepsilon_1+\dots+\varepsilon_p)$ when $1\leq p\leq m-1$, and $\pi_{k,m}$ denotes the sum of the irreducible representations with highest weight $(k\varepsilon_1+\dots+\varepsilon_m)$ and $(k\varepsilon_1+\dots+\varepsilon_{m-1}-\varepsilon_m)$ respectively.

You can compute $\dim \pi_{k,p}$ by using Weyl's dimension formula.

Harmonic Analysis of the de Rham complex of the sphere, Crelles 1989 $\endgroup$ – Donu Arapura Apr 17 '15 at 8:20