I suppose that the standard $3$-simplex must be defined in $\mathbb{R}^4$ because you need four independientes unit vectors.
But, since a $n$-simplex is generate by $n+1$ affinely independient points $p_0,\dots,p_n$, I think that in $\mathbb{R}^3$ you can describe three coplanar points and another one "external" to them, so you get a thetaedron ($3$-simplex) in $\mathbb{R}^3$.
But the standard one is in $\mathbb{R}^4$. How is this possible?
Thanks