As you mentioned, the Skein relation can be understood as R-matrix in the fundamental representation has two eigenvalues: in some normalization $$(R-q)(R+1/q)=0~.$$ In colored cases, the number of eigenvalues increases and one has $$\prod_i (R-\lambda_i(q))=0~,$$ which is no longer quadratic equation and is less convenient to use. For instance, in a certain basis, we have the relation $$(R-q^6)(R+q^2)(R-1)=0~$$ the second symmetric representation.

Eqn. (2.9) and (2.11) of this paper explicitly writes the Skein relations in colored cases. (See also sec 2 of this paper.)

As you mentioned, the Skein relation can be understood as R-matrix in the fundamental representation has two eigenvalues: in some normalization $$(R-q)(R+1/q)=0~.$$ In colored cases, the number of eigenvalues increases and one has $$\prod_i (R-\lambda_i(q))=0~,$$ which is no longer quadratic equation and is less convenient to use. For instance, in a certain basis, we have the relation $$(R-q^6)(R+q^2)(R-1)=0~,$$ for the second symmetric tensor product of the fundamental representation.

Eqn. (2.9) and (2.11) of this paper explicitly writes the Skein relations in colored cases. (See also sec 2 of this paper.)