You can gain a deeper understanding of the *n*−1 term through geometry alone, not just why it's not *n* but why it takes exactly this form, but you may first need to build up your intuition cope with *n*-dimensional geometry. From there, however, it's a small step to a deeper understanding of degrees of freedom in linear models (i.e. model df & residual df). I think there's little doubt that [Fisher][1] thought this way. Here's a book that builds it up gradually:

Saville DJ, Wood GR. *Statistical methods: the geometric approach*. 3rd edition. New York: Springer-Verlag; 1991. 560 pages. [9780387975177][2]

(Yes, 560 pages. I did say gradually.)

  [1]: http://en.wikipedia.org/wiki/Ronald_Fisher
  [2]: http://en.wikipedia.org/wiki/Special:BookSources/9780387975177