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A bilinear map 
$\varPhi :\mathbb {R}^r\times \mathbb {R}^s\to \mathbb {R}^n$ is nonsingular if 
$\varPhi (\overrightarrow {a},\overrightarrow {b})=\overrightarrow {0}$ implies 
$\overrightarrow {a}=\overrightarrow {0}$ or 
$\overrightarrow {b}=\overrightarrow {0}$. These maps are of interest to topologists, and are instrumental for the study of vector bundles over real projective spaces. The main purpose of this paper is to produce examples of such maps in the range 
$24\leqslant r\leqslant 32,\ 24\leqslant s\leqslant 32,$ using the arithmetic of octonions (otherwise known as Cayley numbers) as an effective tool. While previous constructions in lower dimensional cases use ad hoc techniques, our construction follows a systematic procedure and subsumes those techniques into a uniform perspective.
