First of all, let $V$ be a vector space over the field $\mathbb{F}$. It is possible then to show, by Zorn's Lemma that there is a basis for $V$. The main point is that although basis are quite convenient and their existance is something that helps a lot in Linear Algebra, an element $v\in V$ exists and has a meaning independent of any basis.
In truth, introducing a basis is just means to express each $v\in V$ uniquely as linear combination of a certain set of vectors. This produces a representation of $v$, but $v$ itself is independent of the representation, since given two basis we can work with either one of them and switch from one to another.
Now, in Quantum Mechanics if we let $\mathcal{H}$ be the Hilbert space describing the system and let $\left|\psi\right\rangle\in \mathcal{H}$ we can sometimes express $\left|\psi\right\rangle $ in a number of different basis in the same way I said above, since $\mathcal{H}$ is a topological vector space.
In other words we can express a state uniquely as superposition of certain states.
Now, my point is the following: I've already seem people talking about this saying that when we write $\left|\psi\right\rangle$ as the superposition
$$\left|\psi\right\rangle = \sum_{n=1}^{\infty}c_n \left|u_n\right\rangle$$
then a particle on the state $\left|\psi\right\rangle$ is simultaneously in all of the states $\left|u_n\right\rangle$. I think this is also the point of Schrödinger's cat.
Now, this is something that really bothers me. Because when we write such decomposition we are simply expressing $\left|\psi\right\rangle$ in a certain manner which can be convenient. The vector $\left|\psi\right\rangle$ is simply itself regardless of any bases. More than that, we can write it in any other basis we find convenient. In that setting, for me a basis is much more a convenient way to represent a vector than an essential part of what the vector is.
In that setting, what is behind this idea of superposition in Quantum Mechanics? Why people sometimes say that sort of thing, that the particle in the state $\left|\psi\right\rangle$ is simultaneously in all states $\left|u_n\right\rangle$? This makes any sense, considering my point?