The best part of algebraic topology is when you can solve a problem by doodling. In particular, if you stare at this picture with an open mind, you should soon become convinced that the Thom space of the tangent bundle to the n-sphere is, in fact, homeomorphic to the quotient space SⁿxSⁿ/A, where A is the antidiagonal.

The picture illustrates how to map the total space of this tangent bundle (considered as a subspace of SⁿxRⁿ⁺¹) homeomorphically onto the space SⁿxSⁿ-A, by stereographically projecting, from the point -x, each tangent space T_xSⁿ onto Sⁿ-{-x}. The Thom space of the bundle can be identified with the one-point compactification of the total space, and a little point-set topology will convince you that, if X is compact and B is a closed subset, the one-point compactification of X-B is X/B.

QED