By definition, a set of functions $S = \{\phi_1,\phi_2,...\}$ is called an orthogonal system on the interval $[a,b]$ if $\forall\phi_i,\phi_j,i\neq g: (\phi_i,\phi_j)=0$.
Examples:
1) $S_1 = \{1, \cos x, \sin x, \cos (2x), \sin(2x), \dots, \cos (mx), \sin(mx),...\}, m\in\mathbb{N}_0 $ is an orthogonal system on the interval $[0,2\pi]$.
2) $S_2 = \{...,e^{-imx},...,e^{-ix},1,e^{ix},e^{i2x},...,e^{imx},...\}, m \in \mathbb{Z}$ is orthogonal on the interval $[0,2\pi]$
Question: Can we build Fourier series without using the systems $S_1$ or $S_2$ (Roughly speaking, they are the same) but using another orthogonal system, for example, $S_3 = \{1,x,2x,...,mx,...\}$, which is orthogonal on the interval $[-1,1]$?
Yes. The underlying mechanism here is (from a certain point of view) that square-integrable functions on $[0, 2 \pi]$, denoted by $L^2([0, 2 \pi])$, form a Hilbert space $X$. The special thing about $S_1 = \{x \mapsto \sin(n x), x \mapsto \cos(n x) ~|~ n \in \mathbb{N}\}$ and $S_2 = \{x \mapsto e^{i n x} ~|~ n \in \mathbb{N}\}$ is that they form a complete orthonormal system (also called orthonormal basis) on $X$. Explicitly this means that $S$ is orthogonal, all elements of $S$ have norm 1 and that the only $f \in X$ satisfying $\forall g \in S: (f, g) = 0$ is $f = 0$.
If a subset $S$ of some $L^2$ space (no necessarily the $L^2$ space over $[0, 2 \pi]$) is a countable complete orthonormal system then for all $f \in L^2$
$$ f = \sum_{s \in S} (s, f) s $$
This is exactly what you do when doing fourier series over $[0, 2\pi]$. The convergence of the sum is only guaranteed in $L^2$-Norm however.
Examples for other complete orthonormal systems are Legendre Polynomials on $[-1, 1]$ (Note that your set (3) is not orthogonal), $h_n(x) = H_n(x) e^{-x^2/2}$ on $(-\infty, \infty)$, where $H_n$ are Hermite Polynomials and spherical harmonics on the 2-Sphere.