In the paper,
A V Arutyunov, S E Zhukovskii, "Properties of real surjective quadratic mappings", SB MATH, 2016, 207 (9), DOI:10.1070/SM8611
a similar questions were studied. 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.