Calculating the tangent space of an ellipsoid

Question

I want to calculate the tangent space of $$ M^{n - 1} := \left\{ x \in \mathbb R^n: \frac{(x^1)^2}{a_1^2}+...+\frac{(x^n)^2}{a_n^2}=1\right\}. $$ I already proved that this is an $(n-1)$ dimensional manifold. Now as far as I know, we have $$ T_xM^{n-1} =\left\{(x,v)\in T_x\mathbb R^n:\sum_{i=1}^n\frac{x_iv_i}{a_i^2}=0\right\}. $$ Is there another way to write this? It just looks not completed yet.