In Fibre Bundles (3rd Ed.) by Husemoller, the canonical $k$-dimensional bundle $\gamma^n_k$ on $\mathrm{Gr}_k(\mathbb{R}^n)$ is introduced as the subbundle of the product bundle, $\mathrm{Gr}_k(\mathbb{R}^n) \times \mathbb R^n \overset{p}{\to} \mathrm{Gr}_k(\mathbb{R}^n)$ defined by the fact that the total space of $\gamma^n_k$ is all pairs $(V,x) \in \mathrm{Gr}_k(\mathbb{R}^n) \times \mathbb R^n$ such that $x \in V$.
I am having trouble connecting this with the definition of the canonical bundle as the exterior power of a line bundle, and my question is essentially how are both of these related in this case?
Furthermore, is it possible to formulate the general definition of the canonical bundle using Husemoller's explanation, i.e. as a subbundle of a product bundle, rather than exterior powers?
I don't know Husemoller's book, but that usage of "canonical" is nowadays more commonly called "tautological"; that is, the tautological bundle of $k$-planes over the Grassmannian has fibre over $V$ equal to $V$ itself.
The top exterior product of the cotangent bundle is related only by name.