Similar questions were studied in the paper
A V Arutyunov, S E Zhukovskii, "Properties of real surjective quadratic mappings", SB MATH, 2016, 207 (9), DOI:10.1070/SM8611 .
In particular, for $n=3$ there are necessary and sufficient conditions for a quadratic mapping to be surjective. The paper is written in Russian. Probably it will be translated this year. If it is interesting, I can write a short summary.
