For the purpose of this question, let us define the Radon-Hurwitz number $\rho(n)$ to be the maximal dimension of a subspace $W$ of the the real vector space $\mathbb{R}^{n\times n}$ of $n\times n$ matrices, such that $W\subset\mathrm{GL}_n(\mathbb{R})\cup \{0\}$.

Question: Is there an elementary proof of the inequality $\rho(16n) \le \rho(n) + 8$?

Comments:

1. By "elementary" proof I mean one that does not rely, directly or indirectly, on $K$-theory.
2. From the general formula for $\rho(n)$ it is plain that $\rho(16n) = \rho(n) + 8$. My questions is if the inequality can be derived without using the formula, and ideally only using methods from linear algebra and/or elementary topology.
3. Other, similar inequalities could be of interest too, if derived by elementary methods.