Isomorphisms from $K_5$ to points in $\mathbb R^4$

Question

Is there an isomorphism from the complete graph $K_5$ to a triangular solid in $\mathbb R^4$?

For example $K_3$ is trivially mapped to a triangle, $K_4$ may be mapped to a tetrahedron, so what about $K_5$?

Does this mean that for any $K_n$ there exists an isomorphism from $K_n$ to a triangular solid in $\mathbb R^{n-1}$?