On clutching functions

I'm reading Hatcher's "Vector bundles and K-Theory" (version 2.2, November 2017). In chapter 1, section 1.2, he describes how to construct vector bundles with base space a sphere. I can follow his example (that I will state at the bottom for completeness). But somehow I think he is doing more work than is needed for this construction. So to be concrete, my question is: why is he gluing together two hemispheres? Wouldn't it be equivalent to fix an equator (i.e. some $$S^{k-1}$$ in $$S^k$$) and considering a map $$f: S^{k-1} \to GL_n(\mathbb{R})$$, so that the projection from the quotient of $$S^k \times \mathbb{R}^n$$ (realised by identifying $$(x,v)$$ with $$(x,f(x)v)$$ for $$x \in S^{k-1}$$) to $$S^k$$ gives a vector bundle?

Citation of his construction: "Write $$S^k$$ as the union of its upper and lower hemispheres $$D_+^k$$ and $$D_−^k$$ , with $$D_+^k \cap D_−^k =S^{k−1}$$. Given a map $$f : S^{k−1} \to GL_n(\mathbb{R})$$, let $$E_f$$ be the quotient of the disjoint union $$D_+^k \times R^n \cup D_−^k \times R^n$$ obtained by identifying $$(x,v) \in \partial D_-^k \times \mathbb{R}^n$$ with $$(x,f(x)(v)) \in \partial D_+^k \times \mathbb{R}^n$$ . There is then a natural projection $$E_f \to S^k$$ and this is an $$n$$ dimensional vector bundle (...)"