For finite dimensional spaces, the answer is "yes"; this is a consequence of the Gram-Schmidt orthonormalization process: every finite dimensional inner product space over $\mathbb{R}$ or over $\mathbb{C}$ has an orthonormal basis.
Now let $\mathbf{V}$ be an inner product space, and let $\mathbf{v}_1,\ldots,\mathbf{v}_n$ be an orthonormal basis. Then $T\colon\mathbf{V}\to \mathbf{F}^n$ given by $T\mathbf{v}_i = \mathbf{e}_i$ (i.e., $T$ maps each vector in $\mathbf{V}$ to its coordinate vector relative to the orthonormal basis $\mathbf{v}_1,\ldots,\mathbf{v}_n$) is an invertible linear transformation such that for all $\mathbf{x},\mathbf{y}\in\mathbf{V}$, $\langle \mathbf{x},\mathbf{y}\rangle = T\mathbf{x}\cdot T\mathbf{y}$, where the right hand side is the standard dot product on $\mathbf{F}^n$ ($\mathbf{F}=\mathbb{R}$ or $\mathbb{C}$).