The OP wrote $[-1, 1]$ but refers to Hadamard matrices, whose entries are more strongly in $\{ -1, 1 \}$. Assuming that the former is meant, we can do this with the sine transform matrix. Take $$M_{ab} = \sin \frac{\pi a b}{n+1} \ \text{for} 1 \leq a,b \leq n.$$ Then $$M M^T = \frac{n+1}{2} \text{Id}_n$$ so the singular values of $M$ are $\sqrt{\tfrac{n+1}{2}}$.

I imagine that other variants on the discrete Fourier transform would work simmilarly.

The OP wrote $[-1, 1]$ but refers to Hadamard matrices, whose entries are more strongly in $\{ -1, 1 \}$. Assuming that the former is meant, we can do this with the sine transform matrix. Take $$M_{ab} = \sin \frac{\pi a b}{n+1} \ \text{for} 1 \leq a,b \leq n.$$ Then $$M M^T = \frac{n+1}{2} \text{Id}_n$$ so the singular values of $M$ are $\sqrt{\tfrac{n+1}{2}}$.

I imagine that other variants on the discrete Fourier transform would work simmilarly.

If you do want entries in $\{ -1, 1 \}$, this is also possible; see Theorem 1.2 in Approximately Hadamard matrices and Riesz bases in random frames. Thanks to Aleksei Kulikov for making me aware of this paper.