Schur polynomials form a basis for the space of symmetric polynomials

Let $\lambda = (\lambda_1 \geq \lambda_2 \geq ... \geq \lambda_n)$ be a partition of $\lambda$ (or Young diagram). Schur polynomial for given partition is

$$s_\lambda = \frac{\det \begin{pmatrix} x_1^{\lambda_n} & x_2^{\lambda_n} & \ldots & x_n^{\lambda_n} \\ x_1^{\lambda_{n-1}+1} & x_2^{\lambda_{n-1}+1} & \ldots & x_n^{\lambda_{n-1}+1} \\ \vdots & \vdots & \ddots & \vdots \\ x_1^{\lambda_1 + n - 1} & x_2^{\lambda_1 + n - 1} & \ldots & x_n^{\lambda_1 + n - 1} \end{pmatrix}}{\displaystyle{\prod_{1 \leq i < j \leq n}} (x_i - x_j)},$$ where $\displaystyle{\prod_{1 \leq i < j \leq n}} (x_i - x_j)$ is a $\det$ of Vandermonde matrix.

Prove that Schur polynomials form a basis for the space of symmetric polynomials of $n$ variables.

I know that any symmetric polynomial can be represented as sum and product of elementary symmetric polynomials. So is there any way how any elementary symmetric polynomials can be represented in Schur polynomials and thus any symmetric polynomial?

Thanks!

Too long for a comment: It is given e.g. in Mike Zabrocki's Notes, page 75, part 4 that $$s_\mu= \sum_{\lambda \vdash|\mu|}K_{\mu\lambda}m_\lambda,$$ with $$K_{\mu\lambda}$$ being Kostka numbers. Further on page 40, part 3 a table how to convert power, elementary, monomial, homogenous and forgotten symmetric polynomials is presented including inverses.