Something that helped me make sense of the concept of basis vectors was looking at how we represent vectors in mathematical notation. For example, the rigorous way of writing a vector <1, 2, 3> is 1i_hat + 2j_hat + 3k_hat. These i_hat, j_hat, and k_hat symbols are just the standard orthonormal basis vectors for the R^3 space! i_hat being <1, 0, 0>, j_hat being <0, 1, 0> and k_hat being <0, 0, 1>. Pretty interesting stuff.
Something that helped me make sense of the concept of basis vectors was looking at how we represent vectors in mathematical notation. For example, the rigorous way of writing a vector <1, 2, 3> is 1i_hat + 2j_hat + 3k_hat. These i_hat, j_hat, and k_hat symbols are just the standard orthonormal basis vectors for the R^3 space! i_hat being <1, 0, 0>, j_hat being <0, 1, 0> and k_hat being <0, 0, 1>. Pretty interesting stuff.