I figure (I hope!) this will be different in 10 years or so, but right now you'll have to combine many sources if you want to get a mathematical education purely from free/open sources; in particular, you'll probably want to include lecture notes in your reading list. I have seen few one-stop shops like Dummit&Foote or Lang or Rudin in the public domain.
Here is a selection of notes and texts I know about. I am not 100% sure if all of the below sources are slated to stay on the internet indefinitely (as opposed to eventually getting removed when being published as a book -- something which sadly still happens on occasion), but many of them are -- some are actually licensed CC or GPL -- and they are all definitely meant to be downloaded and read. From what I can tell, most of them, if not all, have been written by authors competent in their subjects (to say the least). Are they all reader-friendly and pedagogically valuable? I don't know, but I don't see a reason for them to be less readable than the average printed textbook. It's not like the publishers have figured out the secret of good writing; Lang's textbooks, for example, are famously hard to read due to inaccuracies and sloppy writing, and generally, printed books suffer from restrictions on length, marketing-based editing (van der Waerden's "Modern Algebra" lost one of its most interesting chapters due to the respective theory getting out of fashion), and the difficulty of correcting errors when they are found.
S. Gill Williamson's materials include, in particular, the very well-written (although horribly typed!) Comprehensive Introduction to Linear Algebra by him and Joel G. Broida; combined with the extra materials by Marvin Marcus posted on the same website, these probably cover enough linear algebra for four semesters worth of study. Hefferon provides an introduction to linear algebra that is probably more suitable for the first year, but I suspect that it is an example of the "very introductory" phenomenon the OP is complaining about.
Other materials on Williamson's webpage includes a discrete maths course and a book on combinatorics, both joint with Edward A. Bender (a familiar name if you work in Young tableau theory).
Category theory is particularly well-represented on the internet, maybe because it attracts the same kind of people who work in FOSS. See Awodey's lecture notes, the TAC reprints including, e.g., the Barr-Wells textbook, the Joy of Cats and many other things. Actually, see this list; I am not going to copypaste it in full!
On representation theory, Peter Webb's RepBook and Etingof's maybe-not-quite-introduction. Also lecture notes by Teleman. William Crawley-Boevey has some lecture notes on topics in representations (Wayback Machine).
Peter J. Cameron's notes include a lot of introductory material, but not only.
For elementary abstract algebra, there is Andrew Baker and James S. Milne.
For commutative algebra: Robert B. Ash, Pete L. Clark, Altman/Kleiman. For an advanced topic, Irena Swanson and Craig Huneke's book on Integral closure.
Jean Gallier's notes, including computer science, algebra and differential geometry.
George M. Bergman on universal algebra.
Keith Conrad on various things, sadly a collection of notes rather than a self-contained text -- but perfect to fill gaps in one's knowledge arising from reading too shallow books. Similarly, Jerry Shurman's materials which seem slightly more systematic, and Patrick Morandi's writing which include "An Introduction to Abstract Algebra via Applications".
The writings of Grothendieck, including the EGA texts, which are still regarded as the complete text on algebraic geometry. Contrast this with Ravi Vakil's 793-page notes, which seem to do a much better job at providing motivation.
Notes by Andreas Gathmann (Wayback Machine) and Thomas Markwig (Wayback Machine) on introductory algebra and algebraic geometry.
Charles Weibel's K-book for K-theory.
Hanspeter Kraft on transformation groups (Wayback Machine).
Jacob Lurie puts his books online. Please don't misunderstand "higher algebra" as some sequel to basic abstract algebra, though; this is cutting-edge category theory.
Lecture notes by Bodo Pareigis, some of it in English.
Mark Reeder's class pages include notes. Pierre Schapira (Wayback Machine). Christoph Schweigert. Romyar Sharifi (Wayback Machine). Michael Stoll. William Stein (Wayback Machine).
Richard Stanley's Enumerative Combinatorics, volume 1.
Needless to say, the list is incomplete. I have basically browsed through my folder, probably missing half of the good stuff.
Best Answer
IMHO, a book called "Vector Calculus" by Marsden and Tromba is a good choice. It uses intuition, especially geometrical one, to explain and illustrate various notions in multivaraible calculus. It is not too rigorous as it aims for first year students (and IMHO mainly for Engineer majors), however it is absolutely not a hand-waving book that omits proofs. I think it has the right balance between rigor and geometrical intuition that coincides with the site that you have linked.