In an N dimensional vector space, yes, you can always choose a basis and write vectors down that way, but it turns out to often be a bad way to work. For example, picture a surface. At any point on the surface, there is a plane of vectors tangent to that surface. It's unnatural to describe the vectors there in terms of a 2x1 matrix, since that requires the choice of a basis of the tangent space. And if you try to choose a basis of the tangent space, you'd probably want it to vary continuously from one point on the surface to the next, but that's not possible to do on many surfaces, such as a sphere, thanks to the Hairy Ball Theorem. [1]

[1] https://en.wikipedia.org/wiki/Hairy_ball_theorem