Equivalent representation of the uniform distribution on the nonnegative part of the unit sphere $\mathbb{S}^{n-1}\cap\mathbb{R}_+^n$

Question

As is well known, the uniform distribution on the unit sphere $\mathbb{S}^{n-1}$ is equivalent to the normalized standard Gaussian distribution (cf. this link). That is, if a random vector $\boldsymbol{x}\sim\mathcal{N}(0,\boldsymbol{I}_n)$, then $\frac{\boldsymbol{x}}{\|\boldsymbol{x}\|_2}$ is uniformly distributed on the unit sphere $\mathbb{S}^{n-1}$. My question is, if there is an equivalent representation for the uniform distribution on the nonnegative part of the unit sphere $\mathbb{S}^{n-1}\cap\mathbb{R}_+^n$?